$\underset{}{}$Impact of COVID-19 Lockdowns on House Sparrows:$\underset{}{}$
---
$\underset{}{}$Comparative Study from an Indian Context$\underset{}{}$

Sudeep R. Bapat1,∗ and Sreeranjini T. M.2
---
1Indian Institute of Management Indore
2University of Hyderabad
<EMAIL_ADDRESS>

Abstract: Due to the outbreak of COVID-19, the last couple of years have been
drastic in terms of human behavioural patterns. People were forced to stay at
home for a very long duration because of the strict lockdown measures imposed
by governments all over the globe. India was no exception, wherein the Indian
government imposed several very strict lockdowns all across the country, which
restricted human activities and their social behaviours. However, such
restrictions were seen to have a positive impact on environment and ecology.
In this paper, we aim to study the changes in House Sparrow sightings, as a
result of the lockdowns. Are the house sparrows back? is the question we try
to answer, using appropriate exploratory analysis and statistical modelling.

Keywords: House Sparrow; ecology; COVID-19; human behaviour; detectibility

##  1 Introduction

The “House sparrow”, scientifically known as Passer domesticus is one of the
most commonly found birds across India. Since a very long time, the
relationship between human beings and sparrows has been harmonious. They are
easily observable in windows and balconies of houses, which shows how well
they have adapted with the human lifestyle. A typical house sparrow sighted in
India is as seen in Figure 1. But since the starting of the 21st century,
their population has been threatened and the population curve is also dipping.
Several reasons have been cited for this decline, one of them being the
increasing urbanization and pollution. Organizations like Nature Forever
Society (NFS) have adopted several measures to conserve sparrows like “Common
Bird Monitoring of India”, “World Sparrow Day”, “Project Save Our Sparrows”,
among others. In literature, there have been many research articles published
in recent times, which also suggest a decline in the number of house sparrows.
Some of these include, an article by Sharma and Binner (2020), where the
authors indeed claim a decline in their number, according to a study conducted
by the Indian Council of Agricultural Research (ICAR). Even though the cause
of decline was not known specifically, it is believed that it was due to the
non-availability of nests, due to modernization and urbanization. Another
article is that by Paul (2015), where similar conclusions are drawn.
Narayanappa (2022) conjectured that the house sparrow is on the verge of
decline due to urbanization which leads to the unavailability of suitable
nesting sites. Modak (2017) studied the impact of urbanization on house
sparrow distribution, particularly in the city of Kolkata, India. A few other
related articles which one may refer to are by Ghosh et al. (2010), Chaudhary
(2020) or Deepalakshmi and Salomi (2019).

In this paper, we focus on a related issue, that of studying the effects of
COVID-19 lockdowns on house sparrow counts. As discussed earlier, the pandemic
impacted the entire world drastically. Human behaviours and social
interactions were hampered, mainly because of the strict lockdowns imposed by
governments across the globe. However, such lockdowns were a boon to the
ecology and the environment. The wildlife sightings increased after COVID hit
and especially during lockdowns. One belief is that since the humans were not
allowed to roam much, they could spend more time observing and recording such
sightings. There have already been a lot of literature stating that the
lockdowns indeed impacted the wildlife in a positive way. One may refer to
Basile et al. (2021), where the authors investigated how stay-at-home orders
affected data submitted by birdwatchers in Italy, Spain and the United
Kingdom. Schrimpf et al. (2021) found that counts of several focal bird
species changed in pandemic-altered areas, usually increasing in comparison to
prepandemic era. Seress et al. (2021) assessed changes in reproductive success
of great tits (Parus major) at two urban habitats, due to the COVID pandemic.
Gordo et al. (2021) studied the birds’ response to the population lockdown by
using bird records collected by a citizen science project in northeastern
Spain.

Figure 1. Image provided by eBird (www.ebird.org) and created on [16/04/2015]

The structure of the paper is as follows: Section 2 provides the methodology
used for the analysis, along with a brief description of the dataset. Section
3 covers the statistical analysis and modelling of the sparrow counts using
appropriate count regression models, while Section 4 includes a brief
discussion.

##  2 Methodology

In this paper, we aim to study the effect of the COVID-19 pandemic and hence
the reduction in human noise on the count of house sparrows in the country of
India. In order to do so, we downloaded data on house sparrow count in India
from e-bird, which is one of the largest biodiversity science projects with
more than 100 million bird sightings contributed annually by eBirders around
the world. On January 30th, 2020, India’s first ever COVID case was reported
in the state of Kerala. As the number of COVID cases increased, the Prime
Minister of India announced a nationwide lockdown on the midnight of March
24th, which lasted initially for 21 days. Further, it was extended up to May
31st, 2020. The movement and activities of the citizens got restricted as a
result of this imposition of lockdown. This sudden pause in the human
activities also astonishingly brought down the air and noise pollution levels.
This gave us a rare opportunity to carry out a comparative study about how the
less crowded and noisy cities affected the lives of birds like house sparrows.
One of the assumptions is that the strict “stay at home” orders might have
brought more people into birdwatching and the time spent on such activities
might have increased. Thus, we hypothesized that there could be an increase in
the daily number of observations after the imposition of lockdown.

For this study and analysis, our dataset contains 187898 e-bird house sparrow
checklists from 01/01/2018 to 30/11/2021 collected by bird observers across
India. We tried to analyze the changes in the number of bird counts during
lockdown as compared to that in previous years. We did a yearly comparison of
the daily number of counts from 2018 to 2021, which very well includes the
COVID phase as well. Additionally, we performed a phase-wise comparison during
the COVID period, by focusing on the daily bird count in each phase, where the
phases are formed by dividing the timeline to ‘before lockdown’, ‘during
lockdown’ and ‘after lockdown’. For a better understanding, the following
points outline the phases which we chose, along with the corresponding dates.

  * •

Phase 1: before first lockdown (before 24/03/2020)

  * •

Phase 2: during first lockdown (24/03/2020 to 31/05/2020)

  * •

Phase 3: before second lockdown (01/06/2020 to 04/04/2021)

  * •

Phase 4: during second lockdown (05/04/2021 to 15/06/2021)

  * •

Phase 5: after second lockdown (16/06/2021 to 31/11/2021)

As mentioned before, the checklists had a total of 187898 observations across
29 variables which include observation count, county, state, locality (hotspot
or personal), date of sighting, time of sighting, duration, number of
observers in the group etc. Firstly, as we wanted to focus on the daily bird
counts, we aggregated the different entries within each day to a single
number. Another aspect of study was to assess the trends in sparrow counts
around sunrise (4 am to 7 am) and sunset (5 pm to 7 pm). The belief was that
these two time periods differ from the others in terms of bird sightings, as
the sparrows are more active and less harmed by pollution and human intrusion.
Finally, we try to model the sparrow counts using an appropriate Poisson
regression model to conclude that there was indeed an effect of the lockdowns
on their count. Further results and analysis are presented in the subsequent
sections.

##  3 Statistical Analysis and Modelling

This section provides a wide array of exploratory and statistical techniques
applied on different sets of bird counts, as a comparison and modelling
approach. Before we begin with any detailed analysis, we will provide a
glimpse of the spatial distribution of sparrows in India, according to the
different states. Figure 2 contains side-by-side gradient maps of India
showing the sparrow counts before COVID (before 30/01/2020) and after. One can
note the clear increase in sparrow counts in the same states after the
outbreak of COVID.

|
---|---
Figure 2. Side-by-side maps of India showing the sparow counts before (left)
and after (right) COVID

###  3.1 Year wise comparison

The first aspect is to conduct an appropriate year wise comparison of bird
counts. To begin with, we wanted to assess the trends in daily observation
counts over time. We hence aggregated the daily observation count for each
date from 1st Jan 2018 to 30th Nov 2021, into a single number. Basic boxplots
and histograms revealed the presence of outliers in the data, which were
removed, separately for each year, using the standard approach of judging the
interquartile range (if an observation is 1.5 times the interquartile range
more than the third quartile (Q3) or 1.5 times the interquartile range less
than the first quartile (Q1), it is considered an outlier). Figure 3 contains
side-by-side boxplots, histograms and separate scatterplots showing the
distribution of bird counts for each year from 2018 to 2021. Right away, one
can notice the increase in the count in the years 2020 and 2021, as compared
to the previous ones. Especially the histograms show that the counts become
more symmetric for years 2020 and 2021. This gives a visual affirmation of the
fact that sparrows were sighted more during COVID as compared to before.

|
---|---

Figure 3. Side-by-side boxplots (top left), histograms (top right) and
scatterplots (bottom) for the bird counts

Table 1. Summary variables of bird counts for each year
---
Year | No. of days | Min | $Q_{1}$ | Median | Mean | $Q_{3}$ | Max
$2018$ | $336$ | $131$ | $372$ | $508$ | $566.5$ | $723.5$ | $1365$
$2019$ | $344$ | $222$ | $508$ | $684$ | $740.2$ | $913.2$ | $1629$
$2020$ | $349$ | $558$ | $944$ | $1130$ | $1224$ | $1481$ | $2356$
$2021$ | $305$ | $544$ | $956$ | $1092$ | $1149$ | $1325$ | $2075$

Basic summary variables of the bird counts for each year is also presented in
Table 1 for completeness. One can clearly note that the average bird count has
indeed increased after the outbreak of COVID.

Further, we also conducted a few other comparisons between the bird counts, by
focusing on different time periods. The first comparison was between ‘before
COVID’ and ‘after COVID’ sparrow counts. In order to do so, we considered the
data from 01/03/2019 to 28/02/2020 as ‘before COVID’ and paired it with the
same dates next year namely, 01/03/2020 to 28/02/2021, which was termed as
‘after COVID’. A suitable way to conduct this comparison is through a non-
parametric approach, namely the Wilcoxon signed rank test. This is an ideal
way to compare the locations of two matched samples. The hypotheses to test
here are whether the true location shift equals zero or is less than zero. The
corresponding p-value which was obtained was $<2.2\times 10^{-16}$, and we
were able to conclude that the counts ‘after COVID’ were indeed substantially
more.

Another possible comparison is that by looking at the entire available data
before 30/1/2020, when the first COVID case was discovered in India, and that
after 30/1/2020. Since both the samples can be considered to be independent,
we applied another non-parametric approach namely, the Wilcoxon rank sum test,
This test is capable of finding whether the locations for both the independent
samples are equal or not. The obtained p-value was again $<2.2\times
10^{-16}$, which concludes that the number of sparrow sightings increased
after the very initial COVID outbreak in India. A more visual representation
of this conclusion can be seen from the comparative histograms given in Figure
4. The y-axis gives the frequencies of the corresponding counts. One can
easily note that for the ‘after’ group, higher frequencies correspond to
larger bird counts.

Figure 4. Comparative histograms for the bird counts before 30/1/2020 and
after.

###  3.2 Phase wise comparison

We will now try to outline the differences between bird counts by considering
the phases in terms of several lockdowns which the government of India
imposed, to curb the rising COVID cases. As discussed before, a belief is that
due to the strict lockdowns, the number of sparrows would increase as a result
of decreased human activities and all forms of pollution. We will focus on the
five phases as described in Section 1, which are denoted as ‘before’, ‘lock1’,
‘after1’, ‘lock2’, and ‘after2’ for convenience. A point to note is that the
two lockdowns namely, first and second, coincided with the first and second
COVID waves, which also had a global effect. Now the sample sizes (number of
days) under these different phases won’t be comparable, but we atleast aim to
spot the differences visually. Table 2 contains some basic summary variables
for the bird counts in different phases. One can note that during both the
lockdowns, even though the sample size is much less than other phases, the
bird counts are still comparable to other time periods. This indicates the
human tendency of engaging themselves in other interests, such as bird
watching, during strict lockdowns, when they were not allowed to even step out
of their houses. For a better understanding, Figure 5 contains side-by-side
boxplots and scatterplots of the sparrow counts for the different phases.
These present similar conclusions as was seen from the summary table. An
interesting observation from the scatterplots is that during summers of every
year, the bird counts are seen to be the least. This may correspond to the
fact that people prefer staying at homes during peak summers and hence
amounting to reduced sparrow sightings.

Table 2. Summary variables of bird counts for each phase
---
Phase | No. of days | Min | $Q_{1}$ | Median | Mean | $Q_{3}$ | Max
before | $754$ | $131$ | $449.8$ | $625.5$ | $710.9$ | $906.5$ | $2356$
lock1 | $68$ | $879$ | $1246$ | $1506$ | $1469$ | $1654$ | $2220$
after1 | $277$ | $589$ | $931$ | $1092$ | $1180$ | $1316$ | $2323$
lock2 | $71$ | $865$ | $1014$ | $1090$ | $1142$ | $1234$ | $1584$
after2 | $164$ | $544$ | $879.2$ | $1043$ | $1088.7$ | $1285$ | $2075$

|
---|---
Figure 5. Side-by-side boxplots (left) and scatterplots (right) for the
different phases

Another interesting comparison is that between sparrows sighted around sunrise
(4am to 7am) and sunset (5pm to 7pm) on a given day. This will provide us with
an idea as to how the sparrow sightings are distributed during the two most
sought after time points for engaging in bird watching for any given day. In
order to carry out this comparison, we adopted the following approaches. The
first idea was to consider only the observations recorded around sunrise and
sunset (separately) during the days of first lockdown in 2020 and the same
dates in 2018. We purposefully picked 2018 as opposed to 2019 to ensure
complete absence of any COVID traits. We observed that the mean daily bird
counts at sunrise increased from 163.8 in 2018 to 597.8 in 2020 and that at
sunset increased largely from 79.5 in 2018 to 620.2 in 2020. Figure 6 contains
comparative scatterplots separately for sunrise and sunset, which show a
connection between the sparrow counts for the same dates in 2018 and 2020
(during the first lockdown). One can note that during the first lockdown in
2020, more mass is towards the top of the vertical line, which indicates more
number of higher bird count frequencies. The second idea was to understand the
overall effects of COVID and lockdown. We thus adopted a similar technique
which is seen before, to split the timeline at the date 30/01/2020, which
amounts to ‘before COVID’ and ‘after COVID’ and compared the counts at sunrise
and sunset. Table 3 contains some basic summary variables for this comparative
study. One can clearly note that after the onset of COVID, the sparrow counts
increased, both during sunrise and sunset.

|
---|---
Figure 6. Comparitive scatterplots of bird counts during sunrise (left) and
sunset(right)

Table 3. Summary variables of bird counts during sunrise and sunset
---
Phase | Min | $Q_{1}$ | Median | Mean | $Q_{3}$ | Max
Sunrise
before | $8$ | $100.2$ | $167$ | $220.3$ | $277.2$ | $1386$
after | $29$ | $251$ | $361$ | $403.3$ | $519$ | $1341$
Sunset
before | $8$ | $77$ | $116$ | $150.8$ | $177.8$ | $1386$
after | $29$ | $199$ | $267.5$ | $314.2$ | $367.5$ | $1326$

###  3.3 Statistical modelling

Since the sparrow sightings is a count data, one would expect that a Poisson
model would fit the observations better. In this section, we will try to
implement suitable Poisson regression models, by taking a few related
predictors and hence analysing their impact on the bird count. There have been
similar attempts in the past to model the bird counts using a suitable
regression technique. One may refer to Link et al. (2006), where the authors
discuss about a hierarchical model to applied on Christmas bird counts, or
Cȩlik and Durmuş (2020), where the authors have statistically evaluated bird
populations in Akdoğan Lakes by means of Poisson and negative binomial
regression models. Now, if we focus on the raw data (without daily
aggregation) it is found to contain more zeroes than expected. The excess
zeroes could be generated by some other process than the Poisson process, like
naturally low sparrow count of the location, lack of observer expertise, etc.
Figure 7 contains a simple histogram for the bird counts coming from the raw
data. One can clearly note the significant spike at $0$. In this scenario, a
Zero Inflated Poisson regression (ZIP regression) would perform better,
accounting for the excess zeroes. Firstly, we fitted a standard Poisson
regression model to predict count using two variables namely, “locality type”
and “starting hour” of observation. Here the variable “locality type” refers
to the code which is meant to help define the type of location used, as
participants in eBird can plot specific locations on a map (P or Personal) or
choose existing locations from a map (H or Hotspot).

Figure 7. A simple histogram of bird counts indicating excess zeroes

We then used a ZIP regression model using the same variables for predicting
counts and phase of lockdown, for modelling the excess zeroes. A ‘Vuong’ test
was used to compare the two models. The Vuong test can be employed to compare
predicted probabilities of non-nested models and is generally used in
comparing zero-inflated models with standard models. If the two models were
not different, we would see a large p-value. In our case, the p-value was
small $(<2.2\times 10^{-16})$ which indicated that the ZIP model performed
better than the standard Poisson one. ZIP regression is an example of the type
of statistical model called the ‘latent variable model’, where two or more
processes occur together, which affect the response variable. In our case, we
used the variables locality type and starting hour of observation for
predicting counts and phase of lockdown for modelling the excess zeroes. There
are two parts to this model. The first one is the standard Poisson regression
part which is given by,

$\log(\lambda)=\beta_{0}+\beta_{1}(\text{locality
type})+\beta_{2}(\text{hour}),$

while the other part models the excess zeroes and is given by,

$logit(\alpha)=\beta_{0}+\beta_{1}(\text{locality type})+\beta_{2}(\text{phase
2})+\beta_{3}(\text{phase 3})+\beta_{4}(\text{phase 4})+\beta_{5}(\text{phase
5}),$

where the five phases are as taken in Section 2. Note that we have picked the
base category to be ‘Phase 1’ (before first lockdown). All the results are in
comparison to this base category. All the coefficients of the model in both
count and inflation parts were statistically significant. For interpretation,
we exponentiate all the coefficients of the model. It is observed that for a
given starting hour of observation, a personal locality is expected to cite
1.143 times birds than a hotspot. And for a given locality type, the bird
count increased by a factor of 1.0012 with 1 hour increase in the starting
hour of observation. The variable ‘phase’ was used to predict excess zeroes in
the model. The log odds of observing a true zero in phase 2 is lesser by
-0.54770 compared to phase 1. It is lesser for phases 3, 4 and 5 by -0.42907,
-0.51192 and -0.44459 respectively, compared to phase 1. This means that the
probability of observing true zero in phase 1, i.e., before lockdown is 0.0642
while it is 0.0381, 0.0428, 0.0395 and 0.0421 for the next 4 phases
respectively.

##  4 Discussion

From this analysis one can note that the house sparrow sightings per day have
increased significantly after the imposition of lockdown. The increase in the
daily number of bird counts could be mainly because more people might have
started birdwatching during the lockdown as a new hobby or interest.
Birdwatching might have helped them ease the hardships of COVID and the
lockdown. This is supported by the fact that the count is 1.143 times more in
a personal locality than in a hotspot. Many people who reported might have
done backyard birdwatching from their homes.

Another conclusion is that the sparrows have been more visible due to quieter
cities and towns with fewer human interactions. There have been various
reports on the declining sparrow population in India, and several individuals
and organizations have taken several efforts to bring awareness about the
protection of sparrows. As a result, sparrows might have been reviving.
However, we do not have enough evidence to claim this. Some other reports,
like “The State of India’s Birds 2020” by a group of research and conservation
organizations, observe that the population of house sparrows has been
relatively stable during the past 25 years. Also, the number of sparrows
observed per citing did not significantly increase in the lockdown periods.
From the comparison of sparrow counts at sunrise and sunset, which are the
popular hours of the day for birdwatching, during lockdown in 2020 and the
same dates in 2018, we can see a significant increase. Similar arguments can
also explain this.

The probability of observing true zeroes decreased slightly in all the phases
since lockdown. This also tells us that since lockdown, the number of people
observing and the duration of observation have increased. More sparrows are
spotted in personal localities, indicating that since there were restrictions
on movement, people watched birds more from their homes rather than travelling
to hotspots. We used the categorical variable ‘phase’ to predict excess zeroes
since the excess zeroes might be produced by people observing for a shorter
duration, and lockdown phases might affect the duration of observation. The
lockdown has also provided birdwatchers more free time to observe, the
duration of observation might have increased, and as a result, zero counts
should be lesser. The results supported this argument as the probability of
observing excess zeroes decreased since the lockdown.

## Data Availability

All data used for the analysis in this paper are not publicly shareable, but
can be directly obtained from https://ebird.org/data/download, after making a
suitable request to eBird.

## References

Basile, M., Francesco Russo, L., Giovanni Russo, V., Senese, A. and Bernardo,
N. (2021). Birds seen and not seen during the COVID-19 pandemic: The impact of
lockdown measures on citizen science bird observations, Biological
Conservation.

Cȩlik, E. and Durmuş, A. (2020). Nonlinear regression applications in
modelling over-dispersion of bird populations, Journal of Animal and Plant
Sciences 30(2).

Chaudhary, B. (2020). Home point study of birds and mammals diversity allied
to humans in lockdown of COVID-19 at Bharatpur, Chitwan, Nepal, Open Journal
of Ecology 10(9): 612-631.

Deepalakshmi, S. and Salomi, A. A. (2019). Impact of urbanization on House
sparrow (Passer domesticus) diversity from Erode and Namakkal districts,
Tamilnadu, India, International Journal of Advanced Research in Biological
Sciences) 6(11).

eBird Basic Dataset. Version: EBD_relNov-2021. Cornell Lab of Ornithology,
Ithaca, New York. Nov 2021.

Ghosh, S., Kim, K. H. and Bhattacharya, R. (2010). A survey on house sparrow
population decline at Bandel, West Bengal, India, Journal of the Korean Earth
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Gordo O., Brotons L., Herrando S. and Gargallo, G. (2021). Rapid behavioural
response of urban birds to COVID-19 lockdown, Proceedings of the Royal Society
B: Biological Sciences.

Link, W. A., Sauer, J. R. and Niven, D. K. (2006). A hierarchical model for
regional analysis of population change using Christmas Bird Count data, with
application to the American Black Duck, Condor 108: 13-24.

Modak, B. K. (2015). Impact of urbanization on house sparrow distribution: A
case study from greater Kolkata, India, Proceedings of the Zoological Society
70: 21-27.

Narayanappa, Y., Gautam, A., Mahobiya, K. and Singh, A. (2022). Short
Communication: A pilot-study on the occurrence and probable factors
influencing the population decline of House Sparrow (Passer domesticus) along
an urbanization gradient in Coimbatore district, India, Biodiversitas 23(8).

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Asia Pacific Journal of Research 1(29).

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J., Robinson, B. G., Warrington, M. H., Mahony, N. A., Horn, A. G., Strimas-
Mackey, M., Fahrig, L. and Koper, N. (2021). Reduced human activity during
COVID-19 alters avian land use across North America, Science Advances 7\.

Seress, G., Sándor, K., Vincze, E., Pipoly, I., Bukor, B., Ágh, N. and Liker,
A. (2021). Contrasting effects of the COVID‑19 lockdown on urban birds’
reproductive success in two cities, Scientific Reports.

Sharma, P. and Binner, M. (2020). The decline of population of house sparrow
in India, International Journal of Agricultural Science 5.


# The LG Fibration

Daniel Livschitz, Weiqing Gu
Claremont Graduate University,
Institute of Mathematical Sciences
Harvey Mudd College

$[$gu<EMAIL_ADDRESS>
###### Abstract

Deep Learning has significantly impacted the application of data-to-decision
throughout research and industry, however, many Deep Learning methods lack a
rigorous mathematical foundation, which creates situations where algorithmic
results fail to be practically invertible. In this paper we present a nearly
invertible mapping between $\mathbb{R}^{2^{n}}$ and $\mathbb{R}^{n+1}$ via a
topological connection between $S^{2^{n}-1}$ and $S^{n}$. Throughout the paper
we utilize the algebra of Multicomplex rotation groups and polyspherical
coordinates to define two maps: the first is a contraction from $S^{2^{n}-1}$
to $\displaystyle\bigotimes^{n}_{k=1}SO(2)$, and the second is a projection
from $\displaystyle\bigotimes^{n}_{k=1}SO(2)$ to $S^{n}$, which together form
a composite map with an almost fibration structure that we call the LG
Fibration. In analogy to the generation of Hopf Fibration using Hypercomplex
geometry from $S^{(2n-1)}\mapsto CP^{n}$, our fibration uses Multicomplex
geometry to project $S^{2^{n}-1}$ onto $S^{n}$. Additionally, we investigate
the algebraic properties of the LG Fibration, ultimately deriving a distance
difference function to determine which pairs of vectors have an invariant
inner product under the transformation. The LG Fibration has applications to
Machine Learning and Artificial Intelligence, in analogy to the current
applications of Hopf Fibrations in adaptive UAV control. Furthermore, the
ability to invert the LG Fibration for nearly all elements allows for the
development of Machine Learning algorithms that may avoid the issues of
uncertainty and reproducibility that currently plague contemporary methods.[7]
The primary result of this paper is a novel method of nearly invertible
geometric dimensional reduction from $S^{2^{n}-1}$ to $S^{n}$, which has
applications for further research in both mathematics and Artificial
Intelligence, including but not limited to the fields of homotopy groups of
spheres, algebraic topology, machine learning, and algebraic biology.

##  1 Introduction

Since the discovery of Quaternions by Olinde Rodrigues[2] and Sir William
Hamilton[3] independently in 1840 and 1843, the anti-commutative Division
Algebras of $\mathbb{H}$ and subsequently $\mathbb{O}$ have been the de facto
generalization of the Complex Numbers in mathematical research. In 1877,
Ferdinand Frobenius proved that all Real finite-dimensional associative
division algebras must be isomorphic with either $\mathbb{R}$, $\mathbb{C}$,
or $\mathbb{H}$. By 1923, Hurwitz’s work on finite-dimensional unital real
algebras was posthumously published, and proved that only in dimensions 1, 2,
4, and 8 are the conditions outlined in the Hurwitz Problem satisfied. These
developments in representation theory, topology, and algebra set the stage for
Heinz Hopf to publish his famous “Hopf Fibration” in 1931. This established a
many to one map from $S^{3}$ to $S^{2}$ using $S^{1}$ fibrations. Adam’s
Theorem further generalized the mapping to $S^{7}$ and $S^{15}$ using
fibrations of $S^{3}$ and $S^{7}$ respectively, proving definitively that a
Homotopy of spheres with a Hopf Invariant of 1 can only be found in dimensions
1, 2, 4, and 8. The algebras of $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, and
$\mathbb{O}$ have a pronounced history in the formation of modern algebra,
however their properties are only a part of the greater connection between
generalizations of the Complex Numbers and Homotopy Groups of Spheres in high
dimensions.

The Multicomplex Numbers are a $2^{n}$ dimensional commutative ring that
extend the matrix representation of Complex Numbers to all higher orders of n,
and contain a large number of geometric subgroups. Due to advancements in
modern algebra and number theory, the once untenable zero divisors that
plagued Multicomplex Numbers can now be explicitly defined with relative ease
and separated from the set to form Abelian high dimensional multiplicative
groups. These properties and more have allowed the Bicomplex numbers to find
numerous applications throughout mathematical physics[11] and algebraic
biology[9]. The construction was first studied by Corrado Segre[12], who
researched a commutative 4 dimensional non division ring extension of the
Complex Numbers, now known as the Bicomplex Numbers. This established a method
of generating consecutively higher dimensional commutative ring extensions of
the Bicomplex Numbers. In 1991, Griffith Price formalized the general
algebraic properties of Multicomplex Numbers in his seminal work “An
Introduction to Multicomplex Spaces and Functions”[10]. In this paper, we will
use the abelian rotation groups in $\mathbb{C}_{n}$ to represent
$\displaystyle\bigotimes^{n}_{k=1}SO(2)$, which serves as the link between
$S^{2^{n}-1}$ and $S^{n}$.

We begin section 2 with a review of the Hopf Fibration construction as well as
additional background into their use in $\mathbb{R}^{3}$ spatial rotations. In
sections 2.2- 2.3, we explore the polyspherical coordinate representation of a
hypersphere as well as define the algebraic properties of the Bicomplex
numbers. Throughout section 3 we explore the motivating example of the LG
Fibration from $S^{3}$ to $S^{2}$, starting in 3.1 with the contraction from
$S^{3}$ onto $\displaystyle\bigotimes^{2}_{k=1}SO(2)$. In 3.2 we define the
projection map from $\displaystyle\bigotimes^{2}_{k=1}SO(2)$ onto $S^{2}$, and
define the LG Fibration as the composition of the two mappings. In section 3.3
we explore how to visualize the LG Fibration through an intermediary mapping
between the contraction and the projection. The LG Fibration is fully
generalized in section 4 as we define the contraction from $S^{2^{n}-1}$ to
$\displaystyle\bigotimes^{n}_{k=1}SO(2)$ as well as the second mapping that
projects from $\displaystyle\bigotimes^{n}_{k=1}SO(2)$ to $S^{n}$. The
definition of “Almost Bijective Map” is given in section 4.4 alongside worked
examples of the surjective vs bijective regions on $S^{3}$ and $S^{7}$.
Section 5 explores the applications of the LG Fibration to Machine Learning by
providing an explicit definition for the ”Difference Function”, which can be
used to compute vector pairs that have an invariant inner product under the LG
Fibration. We conclude the paper in Section 6 with a collection of selected
graphs that display a portion of the unique underlying symmetries present in
the LG Fibration.

The goal of this paper is to demonstrate a novel method for mapping
$S^{2^{n}-1}$ to $S^{n}$ that has applications in fields as diverse as
representation theory, algebraic topology, machine learning, and algebraic
biology. The computationally convenient properties of a commutative high
dimensional algebra provide the mathematical foundation for Geometric
Dimensionality Reduction algorithms, generalized N-bit Algebras, as well as
methods of graphing and data visualization that rely on Bicomplex rather than
Quaternionic algebra.

##  2 Background

###  2.1 The Hopf Fibration

The Hopf Fibration was a breakthrough in the field of algebraic topology due
to the discovery of a connection between $S^{3}$ and $S^{2}$ through circular
fibers. By associating a great circle in $S^{3}$ with a point in $S^{2}$, a
geometric mapping with connections to unit quaternions and applications to
spatial rotations was formed. The Hopf Fibration can be defined as the
following:

######  Definition 2.1 (Hopf Fibration).

Let $a^{2}+b^{2}+c^{2}+d^{2}=1$, $a,b,c,d\in\mathbb{R}\\\ H:S^{3}\mapsto
S^{2},\quad H(a,b,c,d)=\Big{(}a^{2}+b^{2}-c^{2}-d^{2},2(ad-bc),2(ac-
bd)\Big{)}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt$ The mapping from $S^{3}$ to
$S^{2}$ can be clearly seen in polyspherical coordinates$\\\ \vskip 3.0pt plus
1.0pt minus
1.0ptH\Big{(}\cos(\theta_{1})\cos(\theta_{3}),\sin(\theta_{1})\cos(\theta_{3}),\cos(\theta_{2})\sin(\theta_{3}),\sin(\theta_{2})\sin(\theta_{3})\Big{)}\\\
=\Big{(}\cos(2\theta_{3}),-\sin(2\theta_{3})\sin(\theta_{1}-\theta_{2}),-\sin(2\theta_{3})\cos(\theta_{1}+\theta_{2})\Big{)}$

Quaternions give a simple way to encode this axis-angle representation in four
numbers, and can be used to apply (calculate) the corresponding rotation to a
position vector $(\mathrm{x},\mathrm{y},\mathrm{z})$, representing a point
relative to the origin in $\mathbf{R}^{3}$ Therefore, a rotation of angle
$\theta$ around the axis defined by the unit vector

$\vec{u}=\left(u_{x},u_{y},u_{z}\right)=u_{x}\mathbf{i}+u_{y}\mathbf{j}+u_{z}\mathbf{k}$

can be represented by a quaternion. This can be done using an extension of
Euler’s formula:

$\mathbf{q}=e^{\frac{\theta}{2}\left(u_{x}\mathbf{i}+u_{y}\mathbf{j}+u_{z}\mathbf{k}\right)}=\cos\frac{\theta}{2}+\left(u_{x}\mathbf{i}+u_{y}\mathbf{j}+u_{z}\mathbf{k}\right)\sin\frac{\theta}{2}$

It can be shown that the desired rotation can be applied to an ordinary vector
$\mathbf{p}=\left(p_{x},p_{y},p_{z}\right)=p_{x}\mathbf{i}+p_{y}\mathbf{j}+p_{z}\mathbf{k}$
in 3-dimensional space, considered as a quaternion with a real coordinate
equal to zero, by evaluating the conjugation of $\mathbf{p}$ by $\mathbf{q}$ :

$\mathbf{p}^{\prime}=\mathbf{q}\mathbf{p}\mathbf{q}^{-1}$

Notice that if $\mathbf{p}$ is defined as a real number, then $\mathbf{p}$
would remain invariant under the conjugation by
$\mathbf{q}\mathbf{p}\mathbf{q}^{-1}$. Recall, using the Hamilton product, for
each unit quaternion we can use it to define a rotation in $R^{3}$ via
conjugation.
$\mathbf{p}^{\prime}=\left(p_{x}^{\prime},p_{y}^{\prime},p_{z}^{\prime}\right)$
is the new position vector of the point after the rotation.

In a programmatic implementation, the conjugation is achieved by constructing
a quaternion whose vector part is $\mathbf{p}$ and real part equals zero, and
then performing the quaternion multiplication. The vector part of the
resulting quaternion is the desired vector $\mathbf{p}^{\prime}$. The key is
that use a unit quaternion number to represent a rotation in $\mathbb{R}^{3}$,
it splits the $\mathbb{R}^{4}$ into $\mathbb{R}$ and $\mathbb{R}^{3}$, which
is the real and imaginary part of $\mathbb{H}$. This is still true for any
other higher $\mathbb{R}^{n}$ which split as into $\mathbb{R}$ and
$\mathbb{R}^{n-1}$ including Cayley numbers or high dimensional Clifford
algebra. The polyspherical coordinates we are going to introduce below allow
us to split $S^{n-1}$ into $S^{n-k-1}(\cos(\theta))$ and
$S^{k-1}(\sin(\theta))$.

###  2.2 Polyspherical Coordinates

In order to understand the techniques derived in this paper, as well as to
heuristically follow the calculations performed, it is essential to discuss a
particular method of generating higher dimensional spherical coordinates
called polyspherical coordinates.

The polyspherical coordinate construction states that the unit sphere in
$\mathbb{R}^{n}$ can be generated inductively using polyspherical coordinates
for $\mathbb{R}^{k}$ and $\mathbb{R}^{n-k}$. The relationship can be stated as
the following:

######  Definition 2.2 (Polyspherical Coordinates).

Let $w\in S^{n-k-1},z\in S^{k-1},\\\ \theta\in[0,2\pi)$

$S^{n-1}\equiv\big{(}w\cos(\theta),z\sin(\theta)\big{)}$

A special case of this construction is when we generalize from
$\mathbb{R}^{n}$ to $\mathbb{R}^{n+1}$ by factoring in a $\cos(\theta_{n})$ to
the previous polyspherical coordinates and adding an independent
$\sin(\theta_{n})$[6]:

$S^{n-1}=\big{\\{}\displaystyle\Big{(}\prod_{k=1}^{n-1}\cos\left(\theta_{k}\right)\Big{)}e_{0}+\sum_{k=1}^{n-1}\Big{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n-1}\cos\left(\theta_{\ell}\right)\Big{)}e_{k}\big{|}$
$\theta_{k}\in[0,\pi),1\leq k<n-1,\theta_{n-1}\in[0,2\pi)\big{\\}}$

$S^{n}\nolinebreak=\big{\\{}\displaystyle\bigg{(}\Big{(}\prod_{k=1}^{n-1}\cos\left(\theta_{k}\right)\Big{)}e_{0}+\sum_{k=1}^{n-1}\Big{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n-1}\cos\left(\theta_{\ell}\right)\Big{)}e_{k}\bigg{)}\cos(\theta_{n})$

$+\sin(\theta_{n})e_{n}\big{|}\quad\theta_{k}\in[0,\pi),1\leq
k<n,\theta_{n}\in[0,2\pi)\big{\\}}$

$=\big{\\{}\displaystyle\Big{(}\prod_{k=1}^{n}\cos\left(\theta_{k}\right)\Big{)}e_{0}+\sum_{k=1}^{n}\Big{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\Big{)}e_{k}\big{|}$

$\theta_{k}\in[0,\pi),1\leq k<n,\theta_{n}\in[0,2\pi)\big{\\}}$

This form of $S^{n}$ will allow us to bridge the connection between
Multicomplex Numbers and polyspherical coordinates. We will now focus on
understanding the mathematics behind the product of rotations in the
Multicomplex Algebra.

###  2.3 Rotations In Bicomplex Space

The Bicomplex numbers contain two unique complex terms i and j such that each
squares to -1. We can use these elements to generate the full basis of the
Bicomplex numbers by first taking the Power Set of the two basis, and then
interpreting set inclusion as multiplication as well as the nullset as 1:

$\mathbb{G}_{2}\equiv\big{\\{}i,j|$ $i^{2}=j^{2}=-1,ij=ji\big{\\}}$

$\mathbb{P}(\mathbb{G}_{2})=\Big{\\{}\\{\emptyset\\},\\{i\\},\\{j\\},\\{i,j\\}\Big{\\}}=\Big{\\{}1,i,j,ij\Big{\\}}$

We can now take the linear combination of the basis set over the Real numbers,
along with the algebraic symmetries of the basis elements, and formally define
a closed form expression for the Bicomplex numbers[10]:

$\mathbb{C}_{2}=\\{a+bi+cj+dij|$
$a,b,c,d\in\mathbb{R},1,i,j,ij\in\mathbb{P}(\mathbb{G}_{2})\\}$

The Bicomplex numbers form a commutative ring, which means that the
multiplication operation for elements in the set is well defined and Abelian.
Furthermore, as a Banach Algebra, maps such as the exponential function are
well defined in $\mathbb{C}_{2}$[10].

$(\mathbb{C}_{2},+,\odot$ $,\|\|,\times)$

If we take the product of two independent rotations over the Bicomplex
Numbers, the resulting space is a closed set that lies on $S^{3}$[4]:

######  Definition 2.3 (Simple $S^{3}$ Rotation Group).

Let $i,j\in\mathbb{G}_{2},\theta\in[0,2\pi),\\\ \psi\in[0,\pi)$

$\mathbb{S}^{3}_{\mathbb{G}_{2}}\equiv\big{\\{}\displaystyle e^{\displaystyle
i\theta}\cdot e^{\displaystyle j\psi}\big{\\}}$
$=\big{\\{}\cos(\theta)\cos(\psi)+i\sin(\theta)\cos(\psi)+j\cos(\theta)\sin(\psi)+ij\sin(\theta)\sin(\psi)\big{\\}}$

We can verify that this surface is confined to $S^{3}$ by taking the inner
product of any possible vector in $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ with
itself:

$w=\displaystyle e^{\displaystyle i\theta}\cdot e^{\displaystyle j\psi}$

$\langle
w,w\rangle=\cos^{2}(\theta)\cos^{2}(\psi)+\sin^{2}(\theta)\cos^{2}(\psi)+\cos^{2}(\theta)\sin^{2}(\psi)+\sin^{2}(\theta)\sin^{2}(\psi)$

$=(\cos^{2}(\theta)+\sin^{2}(\theta))(\cos^{2}(\psi)+\sin^{2}(\psi))=1$

This proves that the magnitude squared is equal to 1, therefore:

$|w|=1$

Another method of factoring the product of two rotations gives insight into
how we generate our projection:

$\displaystyle e^{\displaystyle i\theta}\cdot e^{\displaystyle
j\psi}=(\cos(\theta)+i\sin(\theta))(\cos(\psi)+j\sin(\psi))$

$=(\cos(\theta)+i\sin(\theta))\cos(\psi)+j(\cos(\theta)+i\sin(\theta))\sin(\psi)$

$=\cos(\theta)\cos(\psi)+i\sin(\theta)\cos(\psi)+je^{\displaystyle
i\theta}\sin(\psi)$

##  3 LG Fibration in $\mathbb{R}^{4}$

###  3.1 Mapping $S^{3}$ to $\mathbb{S}^{3}_{\mathbb{G}_{2}}$

In this section we will explore the motivating example for the LG Fibration by
constructing two maps that relate $S^{3}$ with $(S^{1})^{\otimes 2}$, as well
as $(S^{1})^{\otimes 2}$ with $S^{2}$ through $\mathbb{T}^{2}$. The following
diagram shows the relationships between the maps and spaces:

${S_{p}^{3}}$${\mathbb{S}^{3}_{\mathbb{G}_{2}}}$${\mathbb{T}^{2}_{2^{-1}}}$${S^{2}}$$\scriptstyle{f}$$\scriptstyle{h}$$\scriptstyle{\mathbf{P}}$$\scriptstyle{\mu}$

LG$(S_{p}^{3})=\mathbf{P}(f(S_{p}^{3}))=\mu(h(f(S_{p}^{3})))=S^{2}$

polyspherical coordinates allow us to map from $S^{2^{n}-1}$ onto the subspace
$(S^{1})^{\otimes n}$, while the commutative ring structure of Multicomplex
Numbers allow us to represent $(S^{1})^{\otimes n}$ as a rotation groups. We
will define a map from $S^{3}$ onto $S^{2}$, as well as explore the LG
Fibration mapping chain in order to illustrate the similarities and
differences with the Hopf Fibration in low dimensions. The first step in
defining the LG Fibration is establishing a particular coordinate system for
$S^{3}$ to map onto $\mathbb{S}^{3}_{\mathbb{G}_{2}}$:

$S_{p+}^{1}\equiv\\{\big{(}\cos(\theta_{1}),\sin(\theta_{1})\big{)}|$
$\theta_{1}\in[0,\pi)\\}$

$S_{p}^{1}\equiv\\{\big{(}\cos(\theta_{1}),\sin(\theta_{1})\big{)}|$
$\theta_{1}\in[0,2\pi)\\}$

######  Definition 3.1 (Particular $S^{3}$ Coordinate Representation).

Let $\theta_{1}\in[0,\pi),\\\ \theta_{2}\in[0,2\pi),\theta_{3}\in[0,\pi]$

$S_{p}^{3}\equiv\\{\big{(}S_{p+}^{1}\cos(\theta_{3}),S_{p}^{1}\sin(\theta_{3})\big{)}\\}$

$=\\{\big{(}\cos(\theta_{1})\cos(\theta_{3}),\sin(\theta_{1})\cos(\theta_{3}),\cos(\theta_{2})\sin(\theta_{3}),\sin(\theta_{2})\sin(\theta_{3})\big{)}\\}$

Now that we have a particular orientation for $S_{p}^{3}$, we can define a map
that contracts $S_{p}^{3}$ to $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ by grouping
the input variables through their indices:

######  Definition 3.2 (Map From $S_{p}^{3}$ To
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$).

Let $\theta_{1}\in[0,\pi),\theta_{2}\in[0,2\pi),\theta_{3}\in[0,\pi]$

$f(\theta_{1},\theta_{2},\theta_{3})=$

$\\{\theta_{1}\mapsto\theta_{2},$ if $\theta_{3}=\pi$ then
$\theta_{2}\mapsto\theta_{2}\pm\pi,\theta_{3}\mapsto 0\\}$

Using Definition 3.1 and 3.2, we can evaluate explicitly how $S_{p}^{3}$ maps
to $\mathbb{S}^{3}_{\mathbb{G}_{2}}$:

$f\Big{(}\cos(\theta_{1})\cos(\theta_{3}),\sin(\theta_{1})\cos(\theta_{3}),\cos(\theta_{2})\sin(\theta_{3}),\sin(\theta_{2})\sin(\theta_{3})\Big{)}$
$=\big{(}\cos(\theta_{2})\cos(\theta_{3}),\sin(\theta_{2})\cos(\theta_{3}),\cos(\theta_{2})\sin(\theta_{3}),\sin(\theta_{2})\sin(\theta_{3})\big{)}$
$\cong\cos(\theta_{2})\cos(\theta_{3})+i\sin(\theta_{2})\cos(\theta_{3})+j\cos(\theta_{2})\sin(\theta_{3})+ij\sin(\theta_{2})\sin(\theta_{3})$
$=\displaystyle e^{i\theta_{2}}\cdot e^{j\theta_{3}}$

$\theta_{2}\in[0,2\pi),\theta_{3}\in[0,\pi)$

This directly shows how the input of $S_{p}^{3}$ gets mapped onto
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ through $f$:

$f(S_{p}^{3})=\mathbb{S}^{3}_{\mathbb{G}_{2}}$

###  3.2 The Projection From $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ to $S^{2}$

Unlike the Hopf Fibration, which provides a map for all components of the
$S^{3}$ sphere, the LG Fibration preserves 2 of the components of $S^{3}$ in
polyspherical coordinates and integrates the angles of rotation into the
orientation of the reduced components. The factorization given at the end of
section 2.4 provides us with the foundation for a projection from
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ to $S^{2}$ by defining a map on the
$je^{\displaystyle i\theta}\sin(\psi)$ component. Recall, we define
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ as the following:

$\mathbb{S}^{3}_{\mathbb{G}_{2}}\equiv\big{\\{}\displaystyle e^{\displaystyle
i\theta}\cdot e^{\displaystyle j\psi}\big{\\}}$

$=\big{\\{}\cos(\theta)\cos(\psi)+i\sin(\theta)\cos(\psi)+j\cos(\theta)\sin(\psi)+ij\sin(\theta)\sin(\psi)\big{\\}}$

$=\big{\\{}\cos(\theta)\cos(\psi)+i\sin(\theta)\cos(\psi)+je^{\displaystyle
i\theta}\sin(\psi)\big{\\}}$

$\theta_{1}\in[0,2\pi),\theta_{2}\in[0,\pi)$

We can embed $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ in $\mathbb{R}^{4}$ with the
following coordinate representation:

$x=\cos(\theta)\cos(\psi)$

$y=\sin(\theta)\cos(\psi)$

$z=\cos(\theta)\sin(\psi)$

$w=\sin(\theta)\sin(\psi)$

Similarly, we can see that polyspherical Coordinates for $S^{2}$ are defined
as:

$\bar{x}=\cos(\theta)\cos(\psi)$

$\bar{y}=\sin(\theta)\cos(\psi)$

$\bar{z}=\sin(\psi)$

From this we can derive the following relationships:

$\bar{x}=x$

$\bar{y}=y$

$\big{|}\bar{z}\big{|}=\sqrt{z^{2}+w^{2}}$

We can now relate $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ with $S^{2}$ through the
following projection mapping:

$\mathbf{P}\big{(}x+iy+jz+ijw\big{)}$

$=\big{(}x,y,e^{i\big{(}\tan^{-1}(\frac{y}{x})-(\tan^{-1}(\frac{y}{x})mod\pi)\big{)}}\sqrt{z^{2}+w^{2}}\big{)}$

The motivation for this particular shift comes from the Bicomplex
generalization of Euler’s Identity that[8]:

$\displaystyle e^{i\pi}=e^{j\pi}=-1$

If $\theta$ is within the range of $[0,\pi)$, then
$\theta-(\theta\bmod\pi)=0$.

If $\theta$ is within the range of $[\pi,2\pi)$, then
$\theta-(\theta\bmod\pi)=\pi$.

We can use these modular relationships to construct the projection from
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ to $S^{2}$ by mapping $e^{i\theta}$ in
$je^{i\theta}\sin(\psi)$ to $\pm 1$.

$e^{i\theta}\mapsto e^{i(\theta-(\theta\bmod\pi))}$

This implies that the orientation of the z component is dependent on the angle
of rotation $\theta$. We can normalize $\theta-(\theta\bmod\pi)$ by dividing
by $\pi$ to indicate whether the result is 0 or 1. This produces in the
following equality:

$\displaystyle e^{i\left(\theta-(\theta
mod\pi\right))}=(-1)^{\frac{\theta-(\theta mod\pi)}{\pi}}$

This allows us to define the LG Fibration Projection as the map from
$(S^{1})^{\otimes 2}$ embedded in $\mathbb{C}_{2}$ to $S^{2}$:

######  Definition 3.3 (Projection From $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ To
$S^{2}$).

Let $\theta\in[0,2\pi),\psi\in[0,\pi)$

$\displaystyle\mathbf{P}(e^{\displaystyle i\theta}\cdot\vskip 3.0pt plus 1.0pt
minus 1.0pte^{\displaystyle
j\psi})=\mathbf{P}\big{(}\cos(\theta)\cos(\psi)+i\sin(\theta)\cos(\psi)+j\cos(\theta)\sin(\psi)+ij\sin(\theta)\sin(\psi)\big{)}$

$=\mathbf{P}(\cos(\theta)\cos(\psi)+i\sin(\theta)\cos(\psi)+je^{\displaystyle
i\theta}\sin(\psi))$

$=\big{(}\cos(\theta)\cos(\psi),\sin(\theta)\cos(\psi),(-1)^{\frac{(\theta-(\theta
mod\pi))}{\pi}}\sin(\psi)\big{)}$

Using the LG Fibration Projection, we can recover a single cover of $S^{2}$
with half circle fibrations on the poles from
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ as the following:

$x=\cos(\theta)\cos(\psi)$

$y=\sin(\theta)\cos(\psi)$

$z=(-1)^{\frac{(\theta-(\theta mod\pi))}{\pi}}\sin(\psi)$

$\theta\in[0,2\pi),\psi\in[0,\pi)$

$\mathbf{P}(\mathbb{S}^{3}_{\mathbb{G}_{2}})=S^{2}$

###  3.3 Visualization of the LG Fibration in $\mathbb{R}^{3}$

To visualize the points in $S^{2}$ that do not have a unique inverse to
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$, we must first project
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ bijectively onto a surface in
$\mathbb{R}^{3}$. $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ is generated by the
product of a full circle and of half a circle in $\mathbb{C}_{2}$, and is
topologically one-to-one with $\frac{1}{2}$ of a torus in $\mathbb{R}^{3}$.
This relationship is due to the construction of the torus as the product of
two circles embedded in $\mathbb{R}^{3}$. In the given orientation, the half-
circles on the northern and southern hemisphere of the torus are the fibers
that map onto the northern and southern pole on $S^{2}$.

---

We can define the construction of the partial torus in $\mathbb{R}^{3}$ as the
following:

######  Definition 3.4 (Half Torus in $\mathbb{R}^{3}$).

$\theta_{1}\in[0,2\pi),\theta_{2}\in[0,\pi),a_{2}\geq 1$

$\mathbb{T}^{2}_{2^{-1}}\equiv\big{\\{}\Big{(}\cos(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},\sin(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})\Big{)}\big{\\}}$

We can immediately show a one-to-one mapping between
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ and $\mathbb{T}^{2}_{2^{-1}}$ by explicitly
defining a map and as well as inverse mapping through their identical domain:

######  Definition 3.5 (Map from $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ to Half
Torus).

$\theta_{1}\in[0,2\pi),\theta_{2}\in[0,\pi)$

$h(e^{\displaystyle i\theta_{1}}\cdot e^{\displaystyle
j\theta_{2}})=\Big{(}\cos(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},\sin(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})\Big{)}$

$h(\mathbb{S}^{3}_{\mathbb{G}_{2}})=\mathbb{T}^{2}_{2^{-1}}$

All elements in $\mathbb{T}^{2}_{2^{-1}}$ and
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ are uniquely defined by
$\theta_{1},\theta_{2}$, therefore we can simply define the inverse map as the
following:

$h^{-1}\Big{(}\cos(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},\sin(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})\Big{)}\newline
=e^{\displaystyle i\theta_{1}}\cdot e^{\displaystyle j\theta_{2}}$

$\theta_{1}\in[0,2\pi),\theta_{2}\in[0,\pi)$

$h^{-1}(\mathbb{T}^{2}_{2^{-1}})=\mathbb{S}^{3}_{\mathbb{G}_{2}}$

Now that we have defined a bijective map from
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ to $\mathbb{T}^{2}_{2^{-1}}$, we must
complete the relationship shown in the commutative diagram by defining a
function $\mu(\mathbb{T}^{2}_{2^{-1}})$ that maps the partial torus onto
$S^{2}$. This map can be seen as the limit of $\mathbb{T}^{2}_{2^{-1}}$ as
$a_{2}\rightarrow 0.$

######  Definition 3.6 (Map from Half Torus to $S^{2}$).

$\theta_{1}\in[0,2\pi),\theta_{2}\in[0,\pi)$

$\mu\Big{(}\cos(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},\sin(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})\Big{)}$

$=\big{(}\cos(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},\sin(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})\big{)}\newline
\lim a_{2}\rightarrow 0$

$=\big{(}\cos(\theta_{1})\cos(\theta_{2}),\sin(\theta_{1})\cos\theta_{2},(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})\big{)}$

$\mu(\mathbb{T}^{2}_{2^{-1}})=S^{2}$

Now that we have defined
$S^{3}_{p},\mathbb{S}^{3}_{\mathbb{G}_{2}},\mathbb{T}^{2}_{2^{-1}}$, and the
maps between them, we are ready to define the LG Fibration for $S^{3}$. Lets
recall the definition of the contraction and projection mappings:

$f\big{(}S^{3}_{p}\big{)}=\mathbb{S}^{3}_{\mathbb{G}_{2}}$

$\mathbf{P}\big{(}\mathbb{S}^{3}_{\mathbb{G}_{2}}\big{)}=S^{2}$

These relationships allow us to decompose the LG Fibration of $S^{3}$ onto
$S^{2}$ into two mappings, and can be formally defined as the following:

######  Definition 3.7 (LG Fibration for $S^{3}$).

Let $\theta_{1}\in[0,\pi),$ $\theta_{2}\in[0,2\pi),$ $\theta_{3}\in[0,\pi]$
$LG:S_{p}^{3}\mapsto\mathbb{S}^{3}_{\mathbb{G}_{2}}\mapsto S^{2}$
The contraction from $S_{p}^{3}$ to $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ is
achieved by mapping: $\theta_{1}\mapsto\theta_{2}$, and if $\theta_{3}=\pi$,
then $\theta_{3}\mapsto 0$ and $\theta_{2}\mapsto\theta_{2}\pm\pi\\\ $The
projection from $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ to $S^{2}$ is given by
mapping: $je^{i\theta_{2}}\sin(\theta_{3})\mapsto
j(-1)^{\frac{\theta_{2}-(\theta_{2}mod\pi)}{\pi}}\sin(\theta_{3})\\\ \\\
\vskip 3.0pt plus 1.0pt minus
1.0ptLG\Big{(}\cos(\theta_{1})\cos(\theta_{3}),\sin(\theta_{1})\cos(\theta_{3}),\cos(\theta_{2})\sin(\theta_{3}),\sin(\theta_{2})\sin(\theta_{3})\Big{)}\\\
=\mathbf{P}\Big{(}\cos(\theta_{2})\cos(\theta_{3})+i\sin(\theta_{2})\cos(\theta_{3})+j\cos(\theta_{2})\sin(\theta_{3})+ij\sin(\theta_{2})\sin(\theta_{3})\Big{)}\\\
=\Big{(}\cos(\theta_{2})\cos(\theta_{3}),\sin(\theta_{2})\cos(\theta_{3}),(-1)^{\frac{\theta_{2}-(\theta_{2}mod\pi)}{\pi}}\sin(\theta_{3})\Big{)}$

$LG\big{(}S^{3}_{p}\big{)}=\mathbf{P}\big{(}f\big{(}S^{3}_{p}\big{)}\big{)}=S^{2}$

##  4 LG Fibration In High Dimensions

###  4.1 Rotations in Multicomplex Space

Multicomplex numbers allow us to define an Abelian group embedded in
$\mathbb{R}^{2^{n}}$, which is generated by the product of $n$ rotations. The
benefit of a surface produced by $n$ orthogonal rotations in $\mathbb{C}_{n}$,
is that the resulting set is confined to the surface of $S^{2^{n}-1}$ and only
requires $n$ degrees of freedom to represent each point. By performing a
modular reduction on the components of $(S^{1})^{\otimes n}$, the LG Fibration
allows us to reduce vectors from $\mathbb{R}^{2^{n}}$ to $\mathbb{R}^{n+1}$,
as well as explicitly determine the orientation of the reduced components.

We can generalize the construction of Multicomplex Numbers by defining the set
of ”simple” complex units that can be used with the Power set to generate the
full basis for $\mathbb{C}_{n}:$

$\mathbb{G}_{n}=\big{\\{}i_{k}|$ $i^{2}_{k}=-1,i_{m}i_{k}=i_{k}i_{m},1\leq
m,k\leq n\big{\\}}$

$\mathbb{P}\left(\mathbb{G}_{n}\right)\equiv\big{\\{}\\{\emptyset\\},\\{i_{1}\\},...,\\{i_{1},i_{2}\\},...,\\{i_{1},...,i_{n}\\}\big{\\}}$

$=\big{\\{}1,i_{1},...,i_{1}i_{2},...,i_{1}...i_{n}\big{\\}}$

$\mathbb{C}_{n}\equiv\big{\\{}\displaystyle\sum_{u_{k}\in\mathbb{P}(\mathbb{G}_{n})}a_{k}u_{k}|$
$a_{k}\in\mathbb{R},1\leq k\leq 2^{n}\big{\\}}$

Now that we have seen the basic tools required to multiply independent
rotations together, we have the knowledge necessary to derive an explicit
solution for each higher order LG Fibration.

######  Definition 4.1 (Simple $S^{2^{n}-1}$ Rotation Group).

Let $\theta_{1}\in[0,2\pi),\theta_{m}\in[0,\pi),\\\ 1<j\leq n$

$\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}\equiv\big{\\{}\displaystyle\prod^{n}_{k=1}e^{i_{k}\theta_{k}}|$
$i_{k}\in\mathbb{G}_{n}\\}$

$=\big{\\{}\displaystyle\prod_{k=1}^{n}\cos\left(\theta_{k}\right)+\sum_{k=1}^{n}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\right)|$
$i_{k}\in\mathbb{G}_{n}\big{\\}}$

We inductively show the equality of the two definitions of
$\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$ in the following proof:

### Proof of Equality

$2^{1}$ Dimensions

If we set $n=1$, we can observe the following:

$\displaystyle\prod^{1}_{k=1}e^{i_{k}\theta_{k}}=\cos(\theta_{1})+i_{1}\sin(\theta_{1})$

Additionally, we can directly compute the second definition with $n=1$:

$\displaystyle\prod_{k=1}^{1}\cos\left(\theta_{k}\right)+\sum_{k=1}^{1}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=2}^{1}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{0}e^{i_{m}\theta_{m}}\right)=\cos(\theta_{1})+i_{1}\sin(\theta_{1})$

This allows us to conclude the proof of equality for $n=1$:

$\displaystyle\prod^{1}_{k=1}e^{i_{k}\theta_{k}}=\prod_{k=1}^{1}\cos\left(\theta_{k}\right)+\sum_{k=1}^{1}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{1}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\right)$

$2^{n-1}$ Dimensions

For our inductive step, we will assume that the following equality holds for
$n-1$:

$\displaystyle\prod^{n-1}_{k=1}e^{i_{k}\theta_{k}}\displaystyle=\prod_{k=1}^{n-1}\cos\left(\theta_{k}\right)+\sum_{k=1}^{n-1}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n-1}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\right)$

$2^{n}$ Dimensions

We must now show that the equality holds for $n$:

$\displaystyle\prod^{n-1}_{k=1}e^{i_{k}\theta_{k}}\cdot\displaystyle
e^{\displaystyle
i_{n}\theta_{n}}=\displaystyle\prod^{n}_{k=1}e^{i_{k}\theta_{k}}$

$\displaystyle\Big{(}\prod_{k=1}^{n-1}\cos\left(\theta_{k}\right)+\sum_{k=1}^{n-1}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n-1}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\right)\Big{)}\cdot\Big{(}\cos(\theta_{n})+i_{n}\sin(\theta_{n})\Big{)}$

$\displaystyle=\Big{(}\prod_{k=1}^{n-1}\cos\left(\theta_{k}\right)+\sum_{k=1}^{n-1}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n-1}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\right)\Big{)}\cos\left(\theta_{n}\right)+\displaystyle\displaystyle\Big{(}\prod_{k=1}^{n-1}\cos\left(\theta_{k}\right)+\sum_{k=1}^{n-1}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n-1}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\right)\Big{)}i_{n}\sin(\theta_{n})$

$\displaystyle=\prod_{k=1}^{n}\cos\left(\theta_{k}\right)+\sum_{k=1}^{n-1}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\right)+\displaystyle\Big{(}\prod_{m=1}^{n-1}e^{i_{m}\theta_{m}}\Big{)}i_{n}\sin(\theta_{n})$

$\displaystyle=\prod_{k=1}^{n}\cos\left(\theta_{k}\right)+\sum_{k=1}^{n}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\right)$

$\boxed{\displaystyle\prod^{n}_{k=1}e^{i_{k}\theta_{k}}=\prod_{k=1}^{n}\cos\left(\theta_{k}\right)+\sum_{k=1}^{n}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\right)}$

###  4.2 Mapping $S^{2^{n}-1}$ to $\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$

The next step in generalizing the LG Fibration to higher dimensions is
establishing a consistent orientation of $S^{2^{n}-1}$ to map onto
$\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$. We can begin by defining a method of
generating a particular form of polyspherical Coordinates. In section 3.1, we
defined the positive semi-circle $S_{p+}^{1}$ as the following:

$S_{p+}^{1}\equiv\\{\big{(}\cos(\theta_{1}),\sin(\theta_{1})\big{)}|$
$\theta_{1}\in[0,\pi)\\}$

We then utilized the polyspherical structure to define a single cover of
$S_{p}^{3}$ as the following:

$S_{p}^{3}\equiv\\{\big{(}S_{p+}^{1}\cos(\theta_{3}),S_{p}^{1}\sin(\theta_{3})\big{)}|$
$\theta_{3}\in[0,\pi]\\}$

We can generalize the positive hemi-hysphere structure, which would allow us
to define the single cover particular orientation of $S^{2^{n}-1}_{p+}$.

$S^{2^{n}-1}_{p+}\equiv\\{\big{(}S^{2^{n-1}-1}_{p+}\cos(\theta_{2^{n}-1}),S^{2^{n-1}-1}_{p+}\sin(\theta_{2^{n}-1})\big{)}|$
$\theta_{2^{n}-1}\in[0,\pi)\\}$

######  Definition 4.2 (Particular Orientation of $S^{2^{n}-1}$).

Let $\theta_{2^{n}-1}\in[0,\pi]$

$S^{2^{n}-1}_{p}\equiv\\{\big{(}S^{2^{n-1}-1}_{p+}\cos(\theta_{2^{n}-1}),S^{2^{n-1}-1}_{p}\sin(\theta_{2^{n}-1})\big{)}\\}$

### Example Map From $S^{7}$ to $\mathbb{S}^{7}_{\mathbb{G}_{3}}$

Using Definition 4.1 and 4.2, we can define $S^{7}$ and
$\mathbb{S}^{7}_{\mathbb{G}_{3}}$ as well as explictly show how $f$ maps from
$S^{7}$ to $\mathbb{S}^{7}_{\mathbb{G}_{3}}$. This example allows us to
demonstrate how the procedure generalizes for all
$S^{2^{n}-1}\mapsto\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$:

$f(\begin{bmatrix}\cos(\theta_{1})\cos(\theta_{3})\cos(\theta_{7})\\\
\sin(\theta_{1})\cos(\theta_{3})\cos(\theta_{7})\\\
\cos(\theta_{2})\sin(\theta_{3})\cos(\theta_{7})\\\
\sin(\theta_{2})\sin(\theta_{3})\cos(\theta_{7})\\\
\cos(\theta_{4})\cos(\theta_{6})\sin(\theta_{7})\\\
\sin(\theta_{4})\cos(\theta_{6})\sin(\theta_{7})\\\
\cos(\theta_{5})\sin(\theta_{6})\sin(\theta_{7})\\\
\sin(\theta_{5})\sin(\theta_{6})\sin(\theta_{7})\end{bmatrix})=\begin{bmatrix}\cos(\theta_{5})\cos(\theta_{3})\cos(\theta_{7})\\\
\sin(\theta_{5})\cos(\theta_{3})\cos(\theta_{7})\\\
\cos(\theta_{5})\sin(\theta_{3})\cos(\theta_{7})\\\
\sin(\theta_{5})\sin(\theta_{3})\cos(\theta_{7})\\\
\cos(\theta_{5})\cos(\theta_{3})\sin(\theta_{7})\\\
\sin(\theta_{5})\cos(\theta_{3})\sin(\theta_{7})\\\
\cos(\theta_{5})\sin(\theta_{3})\sin(\theta_{7})\\\
\sin(\theta_{5})\sin(\theta_{3})\sin(\theta_{7})\end{bmatrix}$

$\cong\cos(\theta_{5})\cos(\theta_{3})\cos(\theta_{7})+i_{1}\sin(\theta_{5})\cos(\theta_{3})\cos(\theta_{7})+i_{2}\cos(\theta_{5})\sin(\theta_{3})\cos(\theta_{7})+i_{1}i_{2}\sin(\theta_{5})\sin(\theta_{3})\cos(\theta_{7})+i_{3}\cos(\theta_{5})\cos(\theta_{3})\sin(\theta_{7})+i_{1}i_{3}\sin(\theta_{5})\cos(\theta_{3})\sin(\theta_{7})+i_{2}i_{3}\cos(\theta_{5})\sin(\theta_{3})\sin(\theta_{7})+i_{1}i_{2}i_{3}\sin(\theta_{5})\sin(\theta_{3})\sin(\theta_{7})$

$=\displaystyle e^{i_{1}\theta_{5}}\cdot e^{i_{2}\theta_{3}}\cdot
e^{i_{3}\theta_{7}}$

$\theta_{5}\in[0,2\pi),\theta_{3}\in[0,\pi),\theta_{7}\in[0,\pi)$

Now that we have defined the particular orientation of $S^{2^{n}-1}$ that is
in a similar form to $\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$, we must
explicitly define how to complete the map from
$S_{p}^{2^{n}-1}\mapsto\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$. This requires
defining which set of indices for the input variables should be mapped to the
same index in the output. Under our defined orientation of $S_{p}^{2^{n}-1}$,
the input variables of $\theta_{k}$ for $S_{p+}^{2^{n-1}-1}$ are from $1\leq
k\leq 2^{n-1}-1$, while the $\theta_{m}$ for $S_{p}^{2^{n-1}-1}$ are from
$2^{n-1}\leq j\leq 2^{n}-2$. The surface
$\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$ only has $n$ degrees of freedom, so we
must partition the $2^{n}-1$ input variables into n distinct sets:

$\Theta^{2^{m}-1}_{0}=\\{2^{m}-1\\}$

$\Theta^{2^{m}-1}_{\ell}=\\{k+2^{m+\ell-1}-1|$
$k\in\Theta^{2^{m}-1}_{\ell-1}\\}\cup\Theta^{2^{m}-1}_{\ell-1}$

Finally, we can define the specific mappings from $S_{p}^{2^{n}-1}$ to
$\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$ using relations on the indices of the
input variables:

######  Definition 4.3 (Map From $S_{p}^{2^{n}-1}$ to
$\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$).

$f(\theta_{1},...,\theta_{2^{n}-1})=\Bigg{\\{}\begin{array}[]{rcl}\theta_{k}\mapsto\theta_{2^{m}-1}\quad\quad\quad\quad
k\in\Theta^{2^{m}-1}_{n-m},1<m\leq n\\\
\theta_{\ell}\mapsto\theta_{2^{n}-n}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ell\in\Theta^{1}_{n-1}\\\
\theta_{2^{n}-n}\mapsto\theta_{2^{n}-n}\pm\pi,\theta_{2^{n}-1}\mapsto
0\quad\theta_{2^{n}-1}=\pi\end{array}$

$f(S^{2^{n}-1})=\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$

The map $f$ can be heuristically interpreted as grouping vertical columns of
input variables when $S^{2^{n}-1}_{p}$ is written in vector form. Using
Definition 4.3 and the worked example of $S^{7}$, we can explicitly define
$f:S^{2^{n}-1}_{p}\mapsto\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$ as the
following:

$f(S^{2^{n}-1}_{p})=\displaystyle
e^{i_{1}\theta_{2^{n}-n}}\prod_{\ell=2}^{n}e^{i_{\ell}\theta_{2^{\ell}-1}}$

$\theta_{2^{n}-n}\in[0,2\pi),$ $\theta_{2^{\ell}-1}\in[0,\pi),1<\ell\leq n$

###  4.3 General LG Fibration Projection

In section 3.1 we derived the following partial factorization for the product
of $n$ rotations in $\mathbb{C}_{n}$:

$\displaystyle\displaystyle\prod_{k=1}^{n}e^{i_{k}\theta_{k}}=\prod_{k=1}^{n}\cos\left(\theta_{k}\right)+\sum_{k=1}^{n}i_{k}\left(\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\right)$

We can now use this newly derived product form to generalize the LG Fibration
Projection to map from $\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$ to $S^{n}$ by
making the following substitution:

$\displaystyle\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}}\mapsto\prod_{m=1}^{k-1}e^{i_{m}(\theta_{m}-(\theta_{m}\bmod\pi))}$

By applying the generalized LG Fibration Projection to
$\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$, we recover the vector form of
polyspherical coordinates in $\mathbb{R}^{n+1}$ with reduced components scaled
by 1 or -1 depending on if
$\frac{\theta_{k}-\left(\theta_{k}\bmod\pi\right)}{\pi}$ is 0 or 1:

$\mathbf{P}(\displaystyle\prod_{k=1}^{n}e^{\displaystyle
i_{k}\theta_{k}})=\mathbf{P}(\displaystyle\prod_{k=1}^{n}\cos\left(\theta_{k}\right)+\sum_{k=1}^{n}i_{k}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\theta_{m}})$

We can now perform the LG Fibration Projection and map $1=e_{0}$ as well as
$i_{k}=e_{k}$ for the vector representation in $\mathbb{R}^{n+1}$:

$=\displaystyle\bigg{(}\prod_{k=1}^{n}\cos\left(\theta_{k}\right)\bigg{)}e_{0}+\sum_{k=1}^{n}\bigg{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}e^{i_{m}\left(\theta_{m}-(\theta_{m}mod\pi\right))}\bigg{)}e_{k}$

We can rewrite the final product of
$e^{i_{m}\left(\theta_{m}-(\theta_{m}mod\pi\right))}$ as the following:

$\displaystyle
e^{i_{m}\left(\theta_{m}-(\theta_{m}mod\pi\right))}=(-1)^{\frac{\theta_{m}-(\theta_{m}mod\pi)}{\pi}}$

This brings us to the final form for the LG Fibration Projection:

$None$

The result is the vector representation of $S^{2^{n}-1}$ in polyspherical
Coordinates, thus we have proven that the LG Fibration Projection successfully
maps from $\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$ onto $S^{n}$, and can be
seen as the following:

$\mathbf{P}\big{(}\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}\big{)}=S^{n}$

###  4.4 Almost Bijectivity

We define an Almost Bijective Mapping as a map with a measure zero kernal. We
show that the LG Fibration map is Almost Bijective by explictly defining the
subsets for which the uniqueness of the inverse image is not preserved by the
mapping. The generalized LG Fibration is defined as the following:

$\mathbf{P}(\displaystyle\prod_{k=1}^{n}e^{\displaystyle
i_{k}\theta_{k}})\displaystyle=\bigg{(}\displaystyle\prod_{k=1}^{n}\cos\left(\theta_{k}\right)\bigg{)}e_{0}+\sum_{k=1}^{n}\bigg{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}(-1)^{\frac{(\theta_{m}-(\theta_{m}mod\pi))}{\pi}}\bigg{)}e_{k}$

$\theta_{1}\in[0,2\pi),\theta_{k}\in[0,\pi),1<k\leq n$

If we restrict $\theta_{k}$ to $[0,\pi)$, then necessarily $\displaystyle
e^{i_{k}\left(\theta_{k}-(\theta_{k}mod\pi\right))}=1$ for $1<k\leq n$. This
allows us to simplify the mapping as the following:

$\mathbf{P}(\displaystyle\prod_{k=1}^{n}e^{\displaystyle
i_{k}\theta_{k}})\displaystyle=\bigg{(}\displaystyle\prod_{k=1}^{n}\cos\left(\theta_{k}\right)\bigg{)}e_{0}+\bigg{(}\displaystyle\sin\left(\theta_{1}\right)\prod_{k=2}^{n}\cos\left(\theta_{k}\right)\bigg{)}e_{1}+\displaystyle(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sum_{k=2}^{n}\bigg{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\bigg{)}e_{k}$

We can begin by working through the example of
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ mapped Almost Bijectievly onto $S^{2}$, and
understanding for which values the mapping is not unique:

$\displaystyle\mathbf{P}(e^{\displaystyle i_{1}\theta_{1}}\cdot
e^{\displaystyle i_{2}\theta_{2}})=$

$\cos(\theta_{1})\cos(\theta_{2})e_{0}+\sin(\theta_{1})\cos(\theta_{2})e_{1}+(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})e_{2}$

In this coordinate representation of the mapping, $\displaystyle
e^{\displaystyle i_{1}\theta_{1}}$ acts as the horizontal counterclockwise
rotation from the x to the y axis, and $e^{\displaystyle i_{2}\theta_{2}}$
acts as the vertical counterclockwise rotation from the xy plane to the z
axis. We can observe the following for
$\displaystyle\theta_{2}=\frac{\pi}{2}$:

$\displaystyle\mathbf{P}(e^{\displaystyle i_{1}\theta_{1}}\cdot\displaystyle
e^{\displaystyle i_{2}\frac{\pi}{2}})=$

$\cos(\theta_{1})\cos(\frac{\pi}{2})e_{0}+\sin(\theta_{1})\cos(\frac{\pi}{2})e_{1}+(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\frac{\pi}{2})e_{2}$

$=(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}e_{2}$

We see that the following elements defined by $\mathbf{P}(e^{\displaystyle
i_{1}\theta_{1}}\cdot\displaystyle e^{\displaystyle i_{2}\frac{\pi}{2}})$ map
to the same point in $\mathbb{R}^{3}$:

$\mathbf{P}(e^{\displaystyle i_{1}\theta_{1}}\cdot\displaystyle
e^{\displaystyle
i_{2}\frac{\pi}{2}})=\left\\{\begin{array}[]{rcl}e_{2}\quad\quad\theta_{1}\in[0,\pi)\\\
-e_{2}\quad\theta_{1}\in[\pi,2\pi)\end{array}\right.$

We can extend the example to $\mathbb{S}^{7}_{\mathbb{G}_{3}}$ mapped Almost
Bijectievly onto $S^{3}$:

$\displaystyle\mathbf{P}(e^{\displaystyle i_{1}\theta_{1}}\cdot
e^{\displaystyle i_{2}\theta_{2}}\cdot e^{\displaystyle i_{3}\theta_{3}})=$

$\cos(\theta_{1})\cos(\theta_{2})\cos(\theta_{3})e_{0}$

$+\sin(\theta_{1})\cos(\theta_{2})\cos(\theta_{3})e_{1}$

$+(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})\cos(\theta_{3})e_{2}$

$+(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{3})e_{3}$

We can observe a similar effect to the previous example for
$\displaystyle\theta_{3}=\frac{\pi}{2}$:

$\displaystyle\mathbf{P}(e^{\displaystyle i_{1}\theta_{1}}\cdot
e^{\displaystyle i_{2}\theta_{2}}\cdot e^{\displaystyle i_{3}\frac{\pi}{2}})=$

$\cos(\theta_{1})\cos(\theta_{2})\cos(\frac{\pi}{2})e_{0}$

$+\sin(\theta_{1})\cos(\theta_{2})\cos(\frac{\pi}{2})e_{1}$

$+(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})\cos(\frac{\pi}{2})e_{2}$

$+(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\frac{\pi}{2})e_{3}$

$=(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}e_{3}$

$\mathbf{P}(e^{\displaystyle i_{1}\theta_{1}}\cdot e^{\displaystyle
i_{2}\theta_{2}}\cdot e^{\displaystyle
i_{3}\frac{\pi}{2}})=\left\\{\begin{array}[]{rcl}e_{3}\quad\quad\theta_{1}\in[0,\pi)\\\
-e_{3}\quad\theta_{1}\in[\pi,2\pi)\end{array}\right.$

We can observe that if $\theta_{k}=\displaystyle\frac{\pi}{2}$, the components
with a basis of $e_{0}...e_{k-1}$ are all scaled by $\cos(\frac{\pi}{2})=0$.
Due to the map resulting in the same point regardless of $\theta_{m}$ for
$j<k$, this results in a surjective mapping for all elements of the following
form:

$\mathbb{B}_{n}\equiv\\{\displaystyle\prod_{k=1}^{n}e^{\displaystyle
i_{k}\theta_{k}}|$ $\theta_{k}=\displaystyle\frac{\pi}{2},1<k\leq n\\}$

With a closed form expression for the elements that belong to the kernal of
the map, we can explicitly define the inverse mapping for the isomorphism
between $\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}\setminus\mathbb{B}_{n}$ and
$S^{n}$:

$\mathbf{P}^{-1}\big{(}\bigg{(}\displaystyle\prod_{k=1}^{n}\cos\left(\theta_{k}\right)\bigg{)}e_{0}+\sum_{k=1}^{n}\bigg{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\prod_{m=1}^{k-1}(-1)^{\frac{(\theta_{m}-(\theta_{m}mod\pi))}{\pi}}\bigg{)}e_{k}\big{)}\newline
\hphantom{\quad}\quad\quad\quad\quad\quad=\displaystyle\prod_{k=1}^{n}e^{\displaystyle
i_{k}\theta_{k}}\newline \theta_{m}\neq\frac{\pi}{2},1<m\leq n\quad\quad\quad$

Similarly, we can explicitly group all the elements in $\mathbb{B}_{n}$ that
map to the same element in $S^{n}$:

$\mathbf{P}(\displaystyle\prod_{k=1}^{m-1}e^{\displaystyle
i_{k}\theta_{k}}\cdot
e^{\displaystyle\frac{i_{m}\pi}{2}}\cdot\displaystyle\prod_{\ell=m+1}^{n}e^{\displaystyle
i_{\ell}\theta_{\ell}})=\mathbf{P}(\displaystyle
e^{\displaystyle\frac{i_{m}\pi}{2}}\cdot\displaystyle\prod_{\ell=m+1}^{n}e^{\displaystyle
i_{\ell}\theta_{\ell}})$

$\forall$ $\theta_{m}=\frac{\pi}{2},$ $1<m\leq n$

###  4.5 Commutative Diagram

In order to generalize the surjective mapping from a partial torus onto the
entire $S^{n}$, we must explicitly define our construction of a partial torus.
In $\mathbb{R}^{2}$, the surfaces $\mathbb{T}^{1}$ and $S^{1}$ are equivalent,
which means the construction begins to vary when $n>1$. Similar to the
construction of polyspherical coordinates, we can generate
$\mathbb{T}^{n}_{2^{1-n}}$ by scaling $\mathbb{T}^{n-1}_{2^{2-n}}$ with
$\Big{(}a_{n}+cos(\theta_{k})\Big{)}$ and adding an independent
$(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{k})$:

######  Definition 4.4 (Partial Torus in $\mathbb{R}^{n}$).

Let $a_{m}\geq 1,\theta_{1}\in[0,2\pi),\theta_{m}\in[0,\pi),\\\ 1<j\leq n-1.$
We define the partial Torus as:

$\mathbb{T}^{n-1}_{2^{2-n}}\equiv\big{\\{}\displaystyle\bigg{(}\displaystyle\cos\left(\theta_{1}\right)\prod_{k=2}^{n-1}(a_{k}+\cos\left(\theta_{k}\right))\bigg{)}e_{0}+\bigg{(}\displaystyle\sin\left(\theta_{1}\right)\prod_{k=2}^{n-1}(a_{k}+\cos\left(\theta_{k}\right))\bigg{)}e_{1}$

$+\displaystyle(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sum_{k=2}^{n-1}\bigg{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n-1}(a_{\ell}+\cos\left(\theta_{\ell}\right))\bigg{)}e_{k}\big{\\}}$

We can observe how $\mathbb{T}^{n-1}_{2^{2-n}}$ generalizes to
$\mathbb{T}^{n}_{2^{1-n}}$ using the following construction:

$\mathbb{T}^{n}_{2^{1-n}}=\big{\\{}(\mathbb{T}^{n-1}_{2^{2-n}}\big{(}a_{k}+\cos(\theta_{k})\big{)},\sin(\theta_{k}))\big{\\}}$

$=\big{\\{}\displaystyle\bigg{(}\Big{(}\displaystyle\cos\left(\theta_{1}\right)\prod_{k=2}^{n-1}(a_{k}+\cos\left(\theta_{k}\right))\Big{)}e_{0}+\Big{(}\displaystyle\sin\left(\theta_{1}\right)\prod_{k=2}^{n-1}(a_{k}+\cos\left(\theta_{k}\right))\Big{)}e_{1}$

$+\displaystyle(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sum_{k=2}^{n-1}\Big{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n-1}(a_{\ell}+\cos\left(\theta_{\ell}\right))\Big{)}e_{k}\bigg{)}\Big{(}a_{n}+\cos(\theta_{n})\Big{)}$

$+(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{n})e_{n}|$
$a_{m}\geq 1,\theta_{1}\in[0,2\pi),\theta_{k}\in[0,\pi),1<k\leq n\big{\\}}$

$=\big{\\{}\displaystyle\bigg{(}\displaystyle\cos\left(\theta_{1}\right)\prod_{k=2}^{n}(a_{k}+\cos\left(\theta_{k}\right))\bigg{)}e_{0}+\bigg{(}\displaystyle\sin\left(\theta_{1}\right)\prod_{k=2}^{n}(a_{k}+\cos\left(\theta_{k}\right))\bigg{)}e_{1}$

$+\displaystyle(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sum_{k=2}^{n}\bigg{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}(a_{\ell}+\cos\left(\theta_{\ell}\right))\bigg{)}e_{k}|$

$a_{k}\geq 1,\theta_{1}\in[0,2\pi),\theta_{k}\in[0,\pi),1<k\leq n\big{\\}}$

Recall in Section 3.3, we gave the definition for the Map from
$\mathbb{S}^{3}_{\mathbb{G}_{2}}$ to the Half Torus, which is a bijective
mapping between the two spaces. The map can be defined as the following:

$\mathbf{Definition\;3.5}$ (Map from $\mathbb{S}^{3}_{\mathbb{G}_{2}}$ to Half
Torus). Let $\theta_{1}\in[0,2\pi),\theta_{2}\in[0,\pi)$

$h(e^{\displaystyle i\theta_{1}}\cdot e^{\displaystyle
j\theta_{2}})=\Big{(}\cos(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},\sin(\theta_{1})\big{(}a_{2}+\cos(\theta_{2})\big{)},(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})\Big{)}$

$h(\mathbb{S}^{3}_{\mathbb{G}_{2}})=\mathbb{T}^{2}_{2^{-1}}$

It can be trivially verified that all elements in $\mathbb{T}^{n}_{2^{1-n}}$
have a unique representation for all $\theta_{1}\in[0,2\pi)$ and
$\theta\in[0,\pi),1<j\leq n$. This allows us to generalize Definition 3.5 to a
bijective mapping between $\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$ and
$\mathbb{T}^{n}_{2^{1-n}}$:

######  Definition 4.5 (Map from $\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$ to
$\mathbb{T}^{n}_{2^{1-n}}$).

Let $a_{k}\geq 1,$ $\theta_{1}\in[0,2\pi),\\\ \theta_{k}\in[0,\pi),1<k\leq n$

$h(\displaystyle\prod^{n}_{k=1}e^{\displaystyle
i_{k}\theta_{k}})=\bigg{(}\displaystyle\cos\left(\theta_{1}\right)\prod_{k=2}^{n}(a_{k}+\cos\left(\theta_{k}\right))\bigg{)}e_{0}+\bigg{(}\displaystyle\sin\left(\theta_{1}\right)\prod_{k=2}^{n}(a_{k}+\cos\left(\theta_{k}\right))\bigg{)}e_{1}$

$+\;\displaystyle(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sum_{k=2}^{n}\bigg{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}(a_{\ell}+\cos\left(\theta_{\ell}\right))\bigg{)}e_{k}$

The final surjective morphism from $\mathbb{T}^{n}_{2^{1-n}}$ to $S^{n}$ can
be defined as the limit of $\mathbb{T}^{n}_{2^{1-n}}$ as $a_{k}\rightarrow 0$
for all $k$:

######  Definition 4.6 (Map from $\mathbb{T}^{n}_{2^{1-n}}$ to $S^{n}$).

Let $\theta_{1}\in[0,2\pi),\theta_{k}\in[0,\pi),\\\ 1<k\leq n$

$\mu\bigg{(}\Big{(}\displaystyle\cos\left(\theta_{1}\right)\prod_{k=2}^{n}\big{(}a_{k}+\cos\left(\theta_{k}\right)\big{)}\Big{)}e_{0}+\Big{(}\displaystyle\sin\left(\theta_{1}\right)\prod_{k=2}^{n}\big{(}a_{k}+\cos\left(\theta_{k}\right)\big{)}\Big{)}e_{1}$

$+\;\displaystyle(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sum_{k=2}^{n}\Big{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\big{(}a_{\ell}+\cos\left(\theta_{\ell}\right)\big{)}\Big{)}e_{k}\bigg{)}$

$=\bigg{(}\displaystyle\prod_{k=1}^{n}\cos\left(\theta_{k}\right)\bigg{)}e_{0}+\bigg{(}\displaystyle\sin\left(\theta_{1}\right)\prod_{k=2}^{n}\cos\left(\theta_{k}\right)\bigg{)}e_{1}+\displaystyle(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sum_{k=2}^{n}\bigg{(}\sin\left(\theta_{k}\right)\prod_{\ell=k+1}^{n}\cos\left(\theta_{\ell}\right)\bigg{)}e_{k}$

$=\mathbf{P}(\displaystyle\prod^{n}_{k=1}e^{\displaystyle i_{k}\theta_{k}})$

lim $a_{k}\rightarrow 0,$ $1<k\leq n$

The connection between $S_{p}^{2^{n}-1}$,
$\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$, $\mathbb{T}^{n}_{2^{1-n}}$, and
$\mathbb{S}^{n}$ can be expressed through the following diagram:

${S_{p}^{2^{n}-1}}$${\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}}$${\mathbb{T}^{n}_{2^{1-n}}}$${S^{n}}$$\scriptstyle{f}$$\scriptstyle{h}$$\scriptstyle{\mathbf{P}}$$\scriptstyle{\mu}$

LG$(S_{p}^{2^{n}-1})=\mathbf{P}(f(S_{p}^{2^{n}-1}))=\mu(h(f(S_{p}^{2^{n}-1})))=S^{n}$

##  5 Applications of the LG Fibration to Optimal Projections for Machine
Learning

We can now look into properties such as the distance between two vectors
before and after the transformation. In order to do this, we must define the
inner product of two Multicomplex elements:

$\forall
w_{1},w_{2}\in\mathbb{C}_{n}:w_{1}=\displaystyle\prod^{n}_{k=1}e^{i_{k}\alpha_{k}},w_{2}=\displaystyle\prod^{n}_{k=1}e^{i_{k}\beta_{k}}$

$\langle
w_{1},w_{2}\rangle=\langle\displaystyle\prod^{n}_{k=1}e^{i_{k}\alpha_{k}},\displaystyle\prod^{n}_{k=1}e^{i_{k}\beta_{k}}\rangle$

$=\langle\displaystyle\prod^{n}_{k=1}e^{i_{k}(\alpha_{k}-\beta_{k})},1\rangle$

$=\langle\displaystyle\prod_{k=1}^{n}\cos(\alpha_{k}-\beta_{k})+\sum_{k=1}^{n}i_{k}\big{(}\sin(\alpha_{k}-\beta_{k})\prod_{\ell=k+1}^{n}\cos(\alpha_{\ell}-\beta_{\ell})\prod_{m=1}^{k-1}e^{i_{m}(\alpha_{m}-\beta_{m})}\big{)},1\rangle$

$=\langle\displaystyle\big{(}\prod_{k=1}^{n}\cos(\alpha_{k}-\beta_{k})\big{)}e_{0}$

$\displaystyle+\sum_{k=1}^{n}\Big{(}\sin(\alpha_{k}-\beta_{k})\prod_{\ell=k+1}^{n}\cos(\alpha_{\ell}-\beta_{\ell})\prod_{m=1}^{k-1}e^{i_{m}(\alpha_{m}-\beta_{m})}\Big{)}e_{k},e_{0}\rangle$

$=\displaystyle\prod^{n}_{k=1}\cos(\alpha_{k}-\beta_{k})$

This elegant algebraic result comes from observing that the real component of
the product of rotations must be equal to the product of the real component
from each independent rotation group. Furthermore, we use the equivalence that
the inner product between elements in $\mathbb{C}_{n}$ can be interpreted as
the standard inner product between their vector representations in
$\mathbb{R}^{2^{n}}$[5]. We now have the tools required to compare the inner
product between two vectors before and after the LG Fibration Projection:

$w_{1}=e^{\displaystyle i_{1}\alpha_{1}}\cdot e^{\displaystyle
i_{2}\alpha_{2}}$

$\mathbf{P}(w_{1})=\cos(\alpha_{1})\cos(\alpha_{2})e_{0}+\sin(\alpha_{1})\cos(\alpha_{2})e_{1}+(-1)^{\frac{(\alpha_{1}-(\alpha_{1}mod\pi))}{\pi}}\sin(\alpha_{2})e_{2}$

$w_{2}=e^{\displaystyle i_{1}\beta_{1}}\cdot e^{\displaystyle i_{2}\beta_{2}}$

$\mathbf{P}(w_{2})=\cos(\beta_{1})\cos(\beta_{2})e_{0}+\sin(\beta_{1})\cos(\beta_{2})e_{2}+(-1)^{\frac{(\beta_{1}-(\beta_{1}mod\pi))}{\pi}}\sin(\beta_{2})e_{2}$

We can define a Distance Difference function to measure the difference between
the inner product of the vectors before and after the modular transformation:

$D(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})\equiv\big{|}\langle
w_{1},w_{2}\rangle-\langle\mathbf{P}(w_{1}),\mathbf{P}(w_{2})\rangle\big{|}$

$=\Big{|}\cos(\alpha_{1}-\beta_{1})\cos(\alpha_{2}-\beta_{2})-\Big{(}\cos(\alpha_{1})\cos(\alpha_{2})\cos(\beta_{1})\cos(\beta_{2})+\sin(\alpha_{1})\cos(\alpha_{2})\sin(\beta_{1})\cos(\beta_{2})+(-1)^{\frac{(\alpha_{1}-\alpha_{1}mod\pi)}{\pi}}(-1)^{\frac{(\beta_{1}-\beta_{1}mod\pi)}{\pi}}\sin(\alpha_{2})\sin(\beta_{2})\Big{)}\Big{|}$

$=\Big{|}\cos(\alpha_{1}-\beta_{1})\cos(\alpha_{2}-\beta_{2})-\Big{(}\cos(\alpha_{2})\cos(\beta_{2})\big{(}\cos(\alpha_{1})\cos(\beta_{1})+\sin(\alpha_{1})\sin(\beta_{1})\big{)}+(-1)^{\frac{(\alpha_{1}-\alpha_{1}mod\pi)}{\pi}}(-1)^{\frac{(\beta_{1}-\beta_{1}mod\pi)}{\pi}}\sin(\alpha_{2})\sin(\beta_{2})\Big{)}\Big{|}$

$=\Big{|}\cos(\alpha_{1}-\beta_{1})\cos(\alpha_{2}-\beta_{2})-\cos(\alpha_{2})\cos(\beta_{2})\cos(\alpha_{1}-\beta_{1})-(-1)^{\frac{(\alpha_{1}-\alpha_{1}mod\pi)}{\pi}}(-1)^{\frac{(\beta_{1}-\beta_{1}mod\pi)}{\pi}}\sin(\alpha_{2})\sin(\beta_{2})\Big{|}$

$=\Big{|}\cos(\alpha_{1}-\beta_{1})\Big{(}\cos(\alpha_{2}-\beta_{2})-\cos(\alpha_{2})\cos(\beta_{2})\Big{)}-(-1)^{\frac{(\alpha_{1}-\alpha_{1}mod\pi)}{\pi}}(-1)^{\frac{(\beta_{1}-\beta_{1}mod\pi)}{\pi}}\sin(\alpha_{2})\sin(\beta_{2})\Big{|}$

$=\Big{|}\cos(\alpha_{1}-\beta_{1})\sin(\alpha_{2})\sin(\beta_{2})-(-1)^{\frac{(\alpha_{1}+\beta_{1}-\alpha_{1}mod\pi-\beta_{1}mod\pi)}{\pi}}\sin(\alpha_{2})\sin(\beta_{2})\Big{|}$

$=\Big{|}\sin(\alpha_{2})\sin(\beta_{2})\big{(}\cos(\alpha_{1}-\beta_{1})-(-1)^{\frac{(\alpha_{1}+\beta_{1}-\alpha_{1}mod\pi-\beta_{1}mod\pi)}{\pi}}\big{)}\Big{|}$

We can algebraically determine that the inner product between two vectors
before and after the transformation is invariant if:

$\displaystyle\sin(\alpha_{2})\sin(\beta_{2})=0$ or
$\cos(\alpha_{1}-\beta_{1})=(-1)^{\frac{(\alpha_{1}+\beta_{1}-\alpha_{1}mod\pi-\beta_{1}mod\pi)}{\pi}}$

The General Difference function can be defined as the following:

$\boxed{D(\alpha_{1},...,\alpha_{n},\beta_{1},...,\beta_{n})=\big{|}\displaystyle\prod^{n}_{k=1}\cos(\alpha_{k}-\beta_{k})-\langle\mathbf{P}(\displaystyle\prod_{k=1}^{n}e^{\displaystyle
i_{k}\alpha_{k}}),\mathbf{P}(\displaystyle\prod_{k=1}^{n}e^{\displaystyle
i_{k}\beta_{k}})\rangle\big{|}}$

Minimizing the function determines which pairs of vectors have an inner
product that is invariant under the LG Fibration. This means that the distance
between a pair of vectors in the original space as well as in the
dimensionally reduced space is equivalent, which makes the General Difference
function equal to zero. Computationally solving for these regions allows for
the optimal distribution of data for minimal information loss from applying
the LG Fibration.

##  6 Selected Graphs

We can observe parameterizations of $\mathbb{S}^{3}_{\mathbb{G}_{2}}$
projected down onto $S^{2}$, and compare how the projection differs from a
standard parameterization of polyspherical Coordinates for $S^{2}$. The LG
Fibration Projection $\mathbf{P}(e^{\displaystyle i_{1}\theta_{1}}\cdot
e^{\displaystyle i_{2}\theta_{2}})$ takes the following form:

$x^{\prime}=\cos(\theta_{1})\cos(\theta_{2})$

$y^{\prime}=\sin(\theta_{1})\cos(\theta_{2})$

$z^{\prime}=(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{2})$

We can create a parametric line of a projected rotation through the following
constraint:

$\theta_{2}=a\theta_{1}$

$x^{\prime}=\cos(\theta_{1})\cos(a\theta_{1})$

$y^{\prime}=\sin(\theta_{1})\cos(a\theta_{1})$

$z^{\prime}=(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(a\theta_{1})$

The following examples allow us to gain heuristic understanding of how the
periodic behavior of the higher dimensional rotations is preserved by the map:

$\theta_{2}=\theta_{1}$

|
---|---
$x^{\prime}=\cos(\theta_{1})\cos(\theta_{1})$ | $x=\cos(\theta_{1})\cos(\theta_{1})$
$y^{\prime}=\sin(\theta_{1})\cos(\theta_{1})$ | $y=\sin(\theta_{1})\cos(\theta_{1})$
$z^{\prime}=(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(\theta_{1})$ | $z=\sin(\theta_{1})$

$\theta_{2}=2\theta_{1}$

|
---|---
$x^{\prime}=\cos(\theta_{1})\cos(2\theta_{1})$ | $x=\cos(\theta_{1})\cos(2\theta_{1})$
$y^{\prime}=\sin(\theta_{1})\cos(2\theta_{1})$ | $y=\sin(\theta_{1})\cos(2\theta_{1})$
$z^{\prime}=(-1)^{\frac{(\theta_{1}-(\theta_{1}mod\pi))}{\pi}}\sin(2\theta_{1})$ | $z=\sin(2\theta_{1})$

Overlapping the LG Fibration with the polyspherical graph allows us to see the
effect of the modular shift. We can also observe that the mapping produces 1
non-differentiable point if $a$ is odd, and 2 non-differentiable points if $a$
is even.

|
---|---

Finally, the XY cross sections form rose curves with $2a$ rose petals if $a$
is even, and $a$ rose petals if $a$ is odd:

|
---|---
$\theta_{2}=2\theta_{1}$ | $\theta_{2}=5\theta_{1}$

##  7 Conclusion

Our investigation began with the derivation of polyspherical coordinates,
which we used to bridge a link with a subgroup of the general Abelian
Multicomplex Rotation Group. The bijective relationship between
$\mathbb{S}^{2^{n}-1}_{\mathbb{G}_{n}}$ and $\mathbb{T}^{n}_{2^{1-n}}$ allows
for the LG fibration to map surjectively onto $S^{n}$ from $S^{2^{n}-1}$. This
procedure immediately opens up the possibility for two separate areas of
research: $\mathbb{R}^{n+1}$ rotations using commutative $\mathbb{R}^{2^{n}}$
dimensional subgroups in pure mathematics, and Dimensionality reduction
algorithms for sparse high dimensional spaces in theoretical computer science.

Large scale implementation of the LG Fibration in industrial Dimensionality
Reduction requires further research into methods of preconditioning data sets
such that the inital data will optimally lie within the defined manifold.
Additional research may also be conducted into determining other suitable low
dimensional manifolds that may be embedded in high dimensional space, for this
would allow alternative approaches to minimize data loss. Numerous graphing
and visualization topics may build off of this research as well, for the long
history of Quaternions being used to model 3D rotations may be rivaled by the
appropriate application of Bicomplex Numbers.[13]

This paper utilizes a variety of algebraic and geometric properties present in
the Multicomplex numbers to derive its results, however numerous questions
regarding the general group theoretic symmetries of the structure are yet to
be resolved and thus require extensive investigation. Future publications
should aim to utilize the abelian rotation and hyperbolic groups in
$\mathbb{C}_{n}$ for mathematical physics, such as generalizing how the split
complex numbers may be used to perform a Lorentz boost in 1+1 Minkowski
space.[1] Discovering the LG Fibration was made possible because of the clear
connection between geometry and algebra in the Multicomplex numbers, and our
hope through this paper is to inspire further research into the applications
of geometric groups embedded in high dimensional spaces.

## References

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