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Didicosm
Loops Across Space
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Didicosm
A didicosm is a certain kind of three-dimensional space, one of ten
platycosms, the name given by the mathematician John Conway^[1] to a
space that has a finite volume, no boundary, and a flat geometry: the
same geometry locally as ordinary Euclidean space. Four of the ten
platycosms are non-orientable (a property for which the Mobius strip
is the most famous two-dimensional example) which would lead to
serious complications if we lived inside such a space, as it would
allow matter to be transformed into its own mirror image just by
travelling around a loop. The remaining six, including the didicosm,
are potentially candidates for the shape of our own universe, if it
happens to be finite.^[2] (The didicosm is also referred to as a
Hantzsche-Wendt Manifold, named after the mathematicians who
discovered it in the 1930s, but we will use the name that Conway gave
it much later.)
A didicosm can be formed by taking a rectangular prism, as in the
image on the right, and "identifying" its faces, so that when you
move within the prism to any point on its boundary, you re-enter it
at a different point, determined by the way the faces are matched up
with each other.
For the two faces labelled Face A this is done in the simplest way
possible: if you hit the front Face A, you jump to the corresponding
point on the back Face A, and vice versa.
For the two faces labelled Face B, the identification is a little
trickier: when you reach points on these faces, you jump to the other
face while also undergoing a rotation by 180 degrees around a
vertical axis, as indicated by the way the labels are rotated in the
image.
For the face labelled Face C, you jump to a new horizontal coordinate
by adding or subtracting half the length of the face, while your
vertical coordinate is reflected in the horizontal midline of the
face. The same thing happens for Face D.
These different instructions might leave you worrying about what
happens along the edges, and at the vertices, of the prism. At first
sight, the instructions seem to be potentially contradictory. For
example, suppose you hit an edge shared by faces labelled Face A and
Face B; the instructions for Face A tell you to jump to another point
at the same height, while those for Face B tell you to jump between
the top and bottom faces. But just as the two faces labelled Face A
only comprise a single rectangle within the didicosm itself, the four
edges shared between faces labelled Face A and Face B only comprise a
single line segment in the didicosm.
Cylinders along edges
In the image on the left, that single line segment is surrounded by a
red cylinder, which is split into quarters in this view, but is a
continuous object in the didicosm. The cylinders of various colours
here, though they appear divided into halves or quarters, each
surround a single line segment and are continuous objects in the
didicosm.
Similarly, there are really only two distinct points in the didicosm
that correspond to vertices of the rectangular prism, and there are
black and white spheres centred on them in this image.
You might also worry that some features on the boundary of the prism
will appear in the didicosm as defects in its geometry, preventing it
from being locally the same as Euclidean space at some points, or
singling out certain places as special. But this is not the case.
Just as you can form a perfectly flat version of a torus (the surface
of a doughnut) by identifying opposite edges of a square, without
causing any irregularity in the geometry, the same thing happens
here: every point in a didcosm looks the same as any other, with no
trace of the construction from the prism surviving.
Since every path that would take you out of the rectangular prism
just leads you back into it somewhere else, the didicosm itself has
no boundary. But to be clear, this way of describing the topology of
the didicosm should not be taken as implying that it really is a box
sitting inside a larger Euclidean space, and some process has
connected up its faces in this peculiar fashion. It is simply a handy
way for us to visualise how things are connected.
Figures in kaleidoscopic view of didicosm
Suppose you were standing inside a didicosm of a fixed size that was
small enough for light to cross the whole space in a fraction of a
second. (This is in contrast to the situation in cosmology, where the
didicosm would be billions of light years across at present, and
would also be changing size, having been expanding ever since the Big
Bang.) The image on the right shows a small part of what you would
see. This "kaleidoscopic" view is formed by gluing extra copies of
our original rectangular prism to its six faces, and then further
copies to their free faces, and so on; in principal we could fill all
of infinite, three-dimensional Euclidean space with copies of the
original prism this way.
These copies of the prism are not only shifted relative to the
original, they are rotated as necessary to allow their faces to match
up correctly. Note that none of the copies are mirror images of the
original; the figure's raised hand is always their right hand.
Classifying the loops
One way to understand the shape of a space is to study the different
kinds of loops that you can draw within it. Suppose you fix a point P
in the space, and consider every possible continuous loop you can
traverse that starts and ends at P. This will include the loop where
you do nothing, and just stay at P, along with loops where you move a
short distance away from P and then come back, as well as loops that
might cross the whole space one or more times in different directions
before returning to P.
That sounds a bit unwieldy, and in a continuous space there will
always be an infinite number of loops that are very similar, if you
treat the slightest variation in the path you take as giving you a
different loop. But we can shift the focus to more interesting
properties of the loops, by considering two loops to be equivalent if
we can continuously deform one into the other, while keeping their
ends fixed at point P. (This kind of continuous deformation of one
path into another is known as a homotopy.) You can imagine that the
loops are made of some kind of perfectly stretchy, perfectly flexible
cord, so their length and shape can be changed as much as you like,
and the only obstacle to transforming one loop into another comes
from the nature of the space itself. The number of possible
equivalence classes of loops will then be much smaller than the
number of loops, though it can still be infinite.
In Euclidean space, or on the surface of a ball, every loop can be
continuously deformed into one that just stays at P (the so-called
trivial loop). But on a torus (the surface of a doughnut), loops that
wind a different number of times around the torus will belong to
different equivalence classes. In fact, we can classify all the
equivalence classes with a pair of integers, (a, b), that count how
many times the loop winds around the torus in each of two directions,
with positive or negative integers depending on the direction we
took.
Torus homotopy
The easiest way to picture this is not by drawing a three-dimensional
doughnut and looking at its surface, or by drawing a single square
whose opposite edges are identified, but by taking a kaleidoscopic
view of the torus, where we fill up two-dimensional Euclidean space
with copies of the square that represents the torus.
The image on the left shows some loops in the equivalence class of
all loops that travel twice around the torus in one direction, and
once in the other direction, so we classify it as the (2,1) class of
loops. In the kaleidoscopic view, we can simply draw an arrow that
travels the chosen number of squares from the base point P in the
original square to another copy of P. We then translate the pieces of
the arrow that lie in other squares back to the original. This
approach automatically satisfies the need for any path that leaves
the square along one edge to re-enter it at the corresponding
position on the opposite edge. (Here, we have also replicated
everything in all of the squares.)
This example also shows that if we take a (2,0) loop that goes twice
around the torus to the right in this diagram, and then follow it
with a (0,1) loop that goes once around the torus in the upwards
direction, the loop we get by joining those loops can be deformed
into the one we get by first doing a (0,1) loop and then a (2,0)
loop. That might seem too obvious to be worth mentioning, but there
are other spaces where the same kind of rule doesn't hold: combining
the same two loops in a different order can yield different
equivalence classes for the combined loop.
In general, the equivalence classes of loops that start and end at a
chosen point in a space form what is known as the fundamental group
of the space (also called the first homotopy group). A group in
mathematics refers to a set of objects and an operation that takes
two of them and yields another; for example, the positive real
numbers can form a group with the operation of multiplication. A
group also needs an identity: an object that leaves other objects
unchanged, like the number 1 leaves numbers unchanged under
multiplication, and an inverse for every object, which acts like the
reciprocal of a number.
In the fundamental group, the operation on equivalence classes of
loops comes from joining two loops together. If X and Y are two
equivalence classes of loops, we can pick any loops in these classes,
follow one from P to P, and then the other from P to P, and the
combined loop will also take us from P to P. We write the equivalence
class of that combined loop as X Y, and we will adopt the convention
that the rightmost loop, the one in Y, is the one we follow first. It
is important to note that after joining the two loops, creating a
combined loop that passes through P halfway through the journey (as
well as starting and ending at P), we do not need to hold the
midpoint fixed at P when we go on to make the whole equivalence class
by deforming that loop. You can see this in the animation of the
(2,1) equivalence class, where two of the specific loops pass through
a third copy of P, but most of them only start and end at copies of
P.
In the fundamental group, the identity is the equivalence class of
the trivial loop, where we start at P and just stay there. The
inverse of any equivalence class is found by taking one of its loops
and travelling around it in the opposite direction. If you follow any
loop from P to P with the same loop traversed in the opposite
direction, it is not hard to see that you can deform the combined
loop into one that simply stays put at P.
For the torus, the group we get by acting on the equivalence classes
of loops this way is isomorphic to the group we get by taking pairs
of integers and adding them as the group operation, by which we mean
that we can identify all the elements of one group with all the
elements of the other, and all the group operations will yield the
same results. We say:
The fundamental group of the torus is Z x Z.
where Z is the symbol mathematicians use for the set of all integers,
and x here refers to the Cartesian product, which just means taking
pairs of things from the sets we have combined this way. So, we are
restating the claim we made earlier, that the equivalence classes of
loops on the torus can be classified by pairs of integers, (a, b),
but we are also saying that the group operations are:
(c, d) (a, b) = (a+c, b+d)
(a, b)^-1 = (-a, -b)
Having seen how things work for a two-dimensional torus, it will
probably be no surprise to learn that for a three-dimensional torus --
the platycosm you get by identifying all three pairs of opposite
faces of a rectangular prism in the simplest possible way -- the
fundamental group is just Z x Z x Z, the set of all triples of
integers, with addition as the group operation again. (Conway calls
this kind of space a "torocosm," but most people call it a 3-torus.)
The didicosm has a different fundamental group than the 3-torus, but
we can find it by examining the kaleidoscopic view we have already
created. In the case of the 2D torus, the identification of each
square in the kaleidoscopic view with a pair of integers seemed like
an obvious choice, but we need to step back and think about this a
bit more carefully. When we consider a loop within the space itself,
we get an arrow in the kaleidoscope, starting at one copy of P and
ending at another. We can classify these arrows by the number of
squares they cross in each direction, and this pair of integers gives
us a recipe that we can use to copy any square into any other square,
purely by shifting it to a new location -- a geometrical operation
that mathematicians call a translation.
Screw rotations
But for the kaleidoscopic view of the didicosm, we need rotations as
well as translations. It turns out that all the possible ways of
copying the original prism can be generated by some combination of
three "screw rotations": translations along certain axes combined
with rotations around those axes.
The image on the right shows the axes of the three screw rotations as
the white arrows labelled X, Y and Z, with the length of the arrow
giving the distance by which we translate. In each case, the rotation
is by 180 degrees around the axis.
The screw rotation we have called X, and its inverse X^-1, take the
two copies of the Face C label into each other. The screw rotation Z,
and its inverse Z^-1, do the same for the Face B labels. The screw
rotation X repeated twice, written X^2, and its inverse X^-2, do the
job for the Face A labels. And for the Face D labels, we can use Z^-1
Y and Y^-1 Z, where we perform the rightmost operation first.
If we look carefully for algebraic relations between these three
screw rotations, we find:
X = Y^2 X Y^2
Y = X^2 Y X^2
X Y Z = 1
where by 1 we mean the transformation that does nothing, which is the
identity in the group of rotations and translations.
These three equations comprise what is known as a group presentation
for the complete set of transformations that give us the
kaleidoscopic view of the didicosm. Though it is not as simple and
concrete as talking about something like pairs of integers, as we
could with the torus, it is enough to pin down all the algebraic
properties of this group. What it is saying is that the group
consists of any product of powers of any of the two elements X and Y
subject to the fact that we can make the substitutions that these
equations permit. We don't need to include Z, because the last
equation lets us rewrite Z as Y^-1 X^-1.
X^2 Y X^2 deformed to show it equals Y
The fundamental group of the didicosm -- a group of equivalence
classes of loops -- is isomorphic to the group of geometrical
transformations that give us the kaleidoscopic view, with any loop
being identified with the transformation that takes the original base
point P to the copy of P where the loop ends up, in the kaleidoscopic
view.
In the animation on the left, we convert the equation Y = X^2 Y X^2,
which we originally found for screw rotations, into a homotopy
between loops. First, we create a loop by going around the X loop
twice (red then green), the Y loop once (blue), then the X loop twice
again (cyan then magenta). Then we deform that combination of five
loops into a loop that is simply the Y loop.
Despite the presence of equations like this in the group
presentation, which sometimes let us replace a complicated loop with
a much simpler one, it is not hard to see that the fundamental group
of the didicosm is still infinite. For example, all the integer
powers of any single one of the generators, X or Y, will be distinct.
Loops and holes
There is a sense in which the two kinds of loops that go around a
torus exactly once in each of the two possible directions indicate
the presence of two different "holes." If we think of a torus as the
two-dimensional surface of a three-dimensional doughnut -- say by
coating the doughnut with icing and then removing the doughnut itself
-- then going around the torus the long way takes you around the
ordinary hole in the doughnut, while going around the short way takes
you around the hollow space where the doughnut used to be.
Torus loops
In the image on the right, the blue and red loops circle those two
kinds of holes, whereas the green loop does not circle a hole, it
just circles a portion of the torus itself.
But spaces in mathematics and cosmology are not generally sitting
inside any particular larger space this way. If we want to discuss
the intrinsic properties of a space -- which are completely
independent of any embedding -- can we still talk about "holes" of the
kind we have described for the torus?
Here is one possibility: the green loop we drew on the torus is the
boundary of a region within the torus, whereas the blue and red loops
are not. So we could use the existence of a loop that is not the
boundary of any region as an indicator of a "hole." This is similar
to the idea that the fundamental group contains loops that can't be
deformed into the trivial loop, and for our example on the torus it
more or less matches up with it, but as we will see it is not exactly
the same in general.
The mathematics that makes the notion of "loops that aren't
boundaries" precise is known as homology theory. The details of
homology theory can be set up in all kinds of different ways, but for
our purposes we will focus on a version known as simplicial homology.
Don't be put off by the terminology: a simplex is just a point, a
line segment, a triangle, a tetrahedron, or the analogous geometrical
object in any higher dimension. An n-simplex is an n-dimensional set,
so a triangle is a 2-simplex, a line segment is a 1-simplex, and a
tetrahedron is a 3-simplex. When we are talking about Euclidean
geometry, an n-simplex is the convex hull of its set of n+1 vertices:
the smallest convex set that contains those vertices.
When we study an n-dimensional space using simplicial homology, first
we need to chop it up into n-simplices. In the case of a
2-dimensional space that means dividing it into triangles, but the
term triangulation is often used, for any number of dimensions, to
mean a division into n-simplices.
Having divided our space into n-simplices, that also gives us a whole
lot of simplices of lower dimension, all the way down to 0-simplices,
or points. For example, each tetrahedron into which we have divided a
3-dimensional space will have four triangular faces, each of which
has three line segments as its edges, each of which has two vertices
as its endpoints. Of course many of these lower-dimensional simplices
will be shared between the higher-dimensional ones, so we need to
make sure that we only count them once.
If we assign numbers (any way we like) to all the individual
vertices, that gives us a way of deciding when a list of vertices in
any of our n-simplices is in ascending order, and we will use that to
assign unambiguous descriptions to all of them.
Triangulated torus
For example, in the image on the left, we have triangulated a torus
(as a square with opposite edges identified) into 18 triangles, and
we have numbered the nine vertices 0 to 8. This then allows us to
label the 27 edges by their two vertices in ascending order: e.g. [2
5], [4 7] etc.; in the diagram, we have drawn an arrow on each edge
from the lower-numbered vertex to the higher. And the 18 triangles
can also be labelled by listing their vertices in ascending order:
e.g. [4 5 8], [1 6 7] etc.
The goal in homology theory is to construct groups (in the
mathematical sense) that make all our concepts amenable to studying
with the tools of abstract algebra, just as we did with the
equivalence classes of loops in the previous section. To that end, in
simplicial homology we start by constructing groups for all the k
-simplices in our triangulated space, where k goes from 0, for the
vertices, up to n, the dimension of the space itself.
Each such group, which we will call C[k] for k = 0, 1, ... n, has as
its elements all possible "formal sums" of k-simplices in the
triangulation multiplied by arbitrary integer coefficients. For
example, any element of C[0] for our triangulation of the torus will
look like:
c[0] [0] + c[1] [1] + ... + c[8] [8]
where the c[v] are any integers, and [0] ... [8] are labels for the
vertices. Similarly, the elements of C[1] are sums of integer
multiples of the 27 edges, [0 1] ... [7 8], and the elements of C[2]
are sums of integer multiples of the 18 triangles, [0 1 4] ... [5 6
8].
The group operation on these C[k] is addition of these sums, which
amounts to adding the coefficients for each k-simplex; the identity
is the empty sum, and we obtain an inverse by negating all the
coefficients. These groups are obviously commutative, aka abelian: it
makes no difference which element comes first and which comes second
when you add them. In fact, these groups are all isomorphic to Z x Z
x ... Z, the Cartesian product of as many copies of the integers as
there are k-simplices in the triangulation.
The next ingredient in simplicial homology formalises the notion of
taking the boundary of a simplex. If we look at a triangle, say [0 1
4], it is not hard to see that we can obtain all the line segments on
its boundary just by dropping each of the 3 vertices in turn from the
list for the triangle, and leaving the remaining ones to describe the
edge; this gives us [1 4], [0 4] and [0 1].
It turns out to be extremely useful to add a small twist to this
process: we write the boundary we obtain this way as a formal sum of
the edges, but we alternate between the signs we give them. So we
will say:
Boundary of triangle [0 1 4] = [1 4] - [0 4] + [0 1]
Why is this useful? Because if we apply the same recipe to the formal
sum of the edges, we get:
Boundary of edge [1 4] = [4] - [1]
Boundary of edge [0 4] = [4] - [0]
Boundary of edge [0 1] = [1] - [0]
Boundary of (Boundary of triangle [0 1 4]) = ([4] - [1]) - ([4] -
[0]) + ([1] - [0]) = 0
This lets us capture the basic notion in topology that the boundary
of a set has no boundary itself; e.g. the boundary of a triangle or a
disk forms a closed loop with no endpoints.
If we take the boundaries of everything in C[k+1], we will get a set
of elements of C[k] that are all closed loops, or similar objects of
other dimensions, such as closed surfaces. Certainly, none of them
will have any boundaries themselves. We will call this set B[k]; it
contains all the formal sums of k-simplices that correspond to the
boundary of something of the next-highest dimension.
B[k] = the boundaries of everything in C[k+1]
Because of the way we have defined our groups, and the boundary
operation, each B[k] will be a group as well, a subgroup of C[k]. In
this context, that means that we can add elements of B[k] and take
their inverses, and the result will always still be in B[k]. Also the
group identity, 0, of C[k], is also in B[k], since it is formally the
"boundary" of 0 in C[k+1].
Now, while B[k] contains all the boundaries of all formal sums of (k
+1)-simplices, and the way we have defined things, we know for sure
that all the boundaries of elements of B[k] are zero, the converse
need not be true. That is, there might be sums of k-simplices in C[k]
whose boundaries are zero, but which do not lie in B[k]. That is,
they are closed loops (or closed surfaces, etc.) or sums of such
things, that do not correspond to the boundary of any
higher-dimensional element. This is precisely the notion that we set
out to capture.
For example, consider the element of C[1] in our triangulation of the
torus:
y = [3 4] + [4 5] - [3 5]
Boundary of y = [4] - [3] + [5] - [4] - [5] + [3] = 0
The element y is not in B[1]; it cannot be created as the boundary of
anything in C[2].
We will give the name K[k] to all the elements of C[k] whose
boundaries are zero. [In mathematical terminology, K[k] is the kernel
of the boundary homorphism from C[k] to C[k-1].] It probably won't
surprise you to learn that K[k] is a subgroup of C[k], and B[k] in
turn is a subgroup of K[k].
Now, we could just ask the question: what elements, if any, are left
in K[k] if we remove all the elements of B[k]? That would give one
description of the phenomenon we are seeking to describe. But it
turns out to be more useful to do something that is very common in
group theory, when we have a normal subgroup sitting inside a group
and we want to compare the two groups. You can follow the link if you
want to know the definition of this in general, but for abelian
groups, all subgroups are normal, and it will be easy to see that the
construction we are about to perform works in this particular case.
Instead of removing elements of the subgroup from the larger group,
we "factor out" the subgroup, effectively treating all of its
elements as being the identity of a new group, called the factor
group (aka quotient group).
A simple example of forming a factor group is to take the integers Z
under addition, and use the subgroup of, say, all multiples of 7,
which we can write as 7Z. The factor group, Z/(7Z), is a new group
where we treat all multiples of 7 as being zero, and addition is
performed modulo 7; we write this as Z[7]. Effectively, we have
sliced up Z into 7 shifted copies of 7Z, and called them 0,1,2,...6
according to the amount we have added to them, to create our new
factor group with just 7 elements. In group theory, when we slice up
a group into copies of a subgroup, the copies are known as cosets,
and when the subgroup is a normal subgroup, the cosets themselves
form a group.
In simplicial homology, our final goal is to construct the factor
groups:
H[k] = K[k] / B[k]
These H[k] are what are known as the homology groups of our space.
Specifically, the first homology group, H[1], tells us to what extent
there are closed loops in the space that are not boundaries of
two-dimensional regions. If all closed loops were the boundaries of
regions, then we would have B[1] = K[1], i.e. we would be setting all
of K[1] to the identity, 0, when we formed the factor group, and H[1]
would be the trivial group, {0}. For the surface of a sphere or the
Euclidean plane, that is exactly what happens: both the fundamental
group and the first homology group are trivial.
Sometimes there is a relatively simple way to construct the first
homology group, H[1], without calculating B[1] or K[1]. If we already
know the fundamental group of the space, we can just "abelianise" it.
As we mentioned earlier, an abelian group is a group where the order
in which you perform the group operation makes no difference. This
relationship between the first homology group and the first homotopy
group is one of the results of the Hurewicz theorem.
In the case of the torus, the first homology group is Z x Z: the same
as the fundamental group, since the fundamental group is already
abelian. What this means is that there is a distinct copy of B[1]
inside K[1] corresponding to every pair of integers, which in turn
corresponds to adding arbitrary integer multiples of the two
different loops that cross the full width of the torus to each of the
elements of B[1].
We can find the first homology group for the didicosm by taking the
equations from our group presentation for the fundamental group, and
adding the rule that we can reorder any of the terms.
So, in the first homology group we have:
X = Y^2 X Y^2 = Y^4 X
Y^4 = 1
Y = X^2 Y X^2 = X^4 Y
X^4 = 1
This tells us that the entire first homology group is generated by
two elements, X and Y, and we only need to include their powers from
0 to 3. In other words, the group contains just 16 elements, which
are essentially the same as pairs of integers (a, b), where a and b
belong to the set {0,1,2,3} and correspond to the powers of X and Y.
The group operation is addition modulo 4. We say:
The first homology group of the didicosm is Z[4] x Z[4].
It can be shown that any abelian group generated by a finite number
of elements is equivalent to a Cartesian product of a certain number
of copies of the full set of integers, and a finite part like this --
the Cartesian product of a number of copies of Z[r] for various
values of r -- though either ingredient can be missing. As we have
noted, the first homology group of the torus is Z x Z, and there is
no finite part. But in the case of the didicosm, there are no copies
of Z, just the finite part we have described.
What this means, when we think back to our definition of the first
homology group as:
H[1] = K[1] / B[1]
is that B[1] is not all of K[1], but nor is K[1] higher-dimensional
than B[1] in the way it is for the torus. Rather, there are exactly
16 copies of B[1] in K[1], and we obtain them by adding terms of the
form:
a X + b Y
to each of the elements of B[1], where a and b are integers from the
set Z[4] = {0,1,2,3}, and by X and Y we mean elements of C[1] (that
is, sums of edges in our triangulation) that correspond to the
equivalence classes of loops, X and Y, that we originally encountered
in the fundamental group of the didicosm. You might object that in
our triangulation of the didicosm there could be many different
elements of C[1] that belong to the same equivalence classes, but
they will all differ from each other by elements of B[1], so it makes
no difference which ones we choose to add to the whole subgroup B[1].
This poses the following puzzle: traversing a loop from the
equivalence class X in the fundamental group 4 times in a row does
not produce a loop that we can deform into a trivial loop; in the
fundamental group X^4 [?] 1. But the first homology group is telling us
that any multiple of 4 times any corresponding element of C[1] must
lie in B[1], which means it is the result of taking the boundary of
some sum of triangles in C[2].
Noncontractible loop as boundary
The image on the right shows one solution found from a triangulation
of the didicosm. Here, the blue line represents the loop X^4, while
the triangles have been coloured to show various integer
coefficients: yellow=1, orange=2, magenta=3, red=4. The blue line is
part of the boundary of a red surface, while all the other surfaces
manage to meet up in a way that cancels out all remaining boundary
terms!
So, there is a sense in which X^4 does arise as a boundary, without
being contractible to a trivial loop. And the reason this is true for
X^4 but not any lower power of X is that the coefficients on the
triangles cannot be made any smaller: the yellow ones are already
just 1, so if we tried to divide everything by 4 we would be left
with fractional coefficients for some surfaces.
References
[1] "Describing the platycosms" by J. H. Conway and J. P. Rossetti.
(2003). Online at arXiv.org.
[2] "The Hantzsche-Wendt Manifold in Cosmic Topology" by R. Aurich
and S. Lustig. Classical and Quantum Gravity Vol. 31 Number 16
(2014). Online at arXiv.org.
[3] "On the coverings of the Hantzsche-Wendt Manifold" by G.
Chelnokov and A. Mednykh. Tohoku Mathematical Journal (2) 74(2):
313-327 (2022). Online at arXiv.org.
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