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Aesthetic Apprehension of Continuum Polyhedral Denizens

by Junior Fellow Tsianina Skyplume
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An analysis of the polyhedral denizens of the Continuum according to the
principles of Aesthetic Appreciation.
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Table of Contents

2 - Statement of Inquiry
3 - Catalog of Denizens
4 - Geometrical Analysis
5 - Structural Principles
6 - Harmonic Resonance
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Bibliography:
- Incabulos Oubliette. "Continuum: Metaphysics", Library of Universal
Knowledge, Hallifax, 262 CE.
- Nejii Talnara. "Studies on Harmony", Arboreal Library, Serenwilde,
256 CE.
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Statement of Inquiry
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While much has been written about the Master Crystals and the structure of
Continuum, most notably by Castellan Incabulos Oubliette in his seminal work
"Continuum: Metaphysics," little is known and less is written concerning the
polyhedral denizens of that plane. Apart from their role in helping to produce
power for the Matrix, they are rarely discussed. Even Incabulos writes only
this:

Being a dead plane, the Continuum can not be said to have any true
'living' denizens. We do however observe a number of polyhedral
entities, though it is impossible to ascertain whether or not they are
sentient. They are mobile and will respond if attacked, though they do
not speak or respond if interacted with in any other way. Since we
cannot say they are sentient, it is reasonable to assume that they
possess no autonomy, and therefore exist as a defensive mechanism for
the plane. It is notable that various species of polyhedron exist and
the number of facets any individual one has will correspond to its
strength.

No academic treatment of their structure, origin, or function has been
published. Lacking historical sources, it is unlikely that the origin of these
crystalline denizens can be definitively determined; however, an analysis of the
structure of these entities, informed by the principles of Aesthetic
Apprehension (and to a lesser extent by the other Abstract Principles), will
shed considerable light on these crystals, why they take the forms they do, and
even provide a compelling theory concerning their origin by which their
structure and creation are mutually caused.
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Catalog of Denizens
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Apart from the Master Crystals and those entities which arrive from elsewhere,
such as Hallifaxian guardians, the only creatures native to the Continuum are
the five polyhedral shapes: the tetrahedron, hexahedron, octahedron,
dodecahedron, and icosahedron. Each of these is a large crystal which possesses
sentience -- that is, awareness of its surroundings and ability to react to them
-- but probably not sapience, an actual understanding and intelligence, though
of this we cannot be sure. Apart from the hexahedral crystals, they move on
their own accord, and all of them respond when provoked by an attack, but do not
respond otherwise.[1]

Each of these five shapes is a crystalline structure containing some form of
energy, or in vernacular, a "spark" within. At a casual glance, the most
noticeable trait of each crystal is that spark, which takes varying forms, and
in some cases seems to influence the sensation the crystal conveys to its
surroundings. After some consideration of the appearances of these crystals,
one might be tempted to postulate an equivalence to the five elements of
kepheran studies of Harmony[2], and indeed, the etymological similarity between
Harmony and Harmonics speaks to a fundamental similarity. However, while this
equivalence might seem compelling on first blush, it is of dubious value and may
speak only to coincidence, or to some deeper and more broadly applicable pattern
of creation.

Tetrahedron: With a pointed four-faceted structure, this crystal already
suggests the element of fire in its flame-like shape and the sharpness of its
vertices. Within it, a red-hued energy is similarly evocative of flame, and
even casts a light which, refracted by the crystalline planes of the
tetrahedron, resembles the flickering of a candle or campfire. As the element
of fire is that of dynamism, it is unsurprising that the tetrahedron moves
swiftly and somewhat erratically.

Hexahedron: Bulkier than its number of facets might otherwise suggest, the
hexahedron barely rises above the ground and does not move appreciably, apart
from a slow rotation. Its dark, blue-tinged shade barely allows any glimmer of
light from within to be seen. Clearly conveying an impression of solidity and
strength, it seems best to suit the element of metal, and its colouration agrees
somewhat, since the silvery sheen of metals often takes on a bluish tone.

Octahedron: Its eight-sided shape filled with a watery mist which dances in
apparently unpredictable patterns, yet ones which seem to be deterministic when
more closely analyzed, the octahedron also aligns itself in perfect balance with
its surroundings. The association with water is obvious, and the lack of any
particular colour to the light within the octahedron corresponds with water's
colourless quality.

Dodecahedron: It is easy to miss the presence of a dodecahedron amongst the air
since its facets, though crystalline, are so evanescent as to appear to merge
with the air that buffets the shape in seemingly random directions. Its only
distinguish feature, apart from a hum of power, is a hint of the azure of the
sky, itself a property of air, further making the association with the element
evident.

Icosahedron: The structure of this twenty-sided shape is far more complex than
the others, and in that complexity demonstrates the gradual approach the shapes
make towards the sphere, suggesting that the Master Crystals might represent the
final stage of evolution of these crystalline shapes. There is nothing evident,
apart from this complexity that is reminiscent of the comparable complexity of
the organic, to associate the icosahedron with the final element of wood.

It is in this more than anything else that the equivalence tends to fall apart.
Is this any more than the mind's tendency to create equivalencies and then force
things to fit when they don't fit on their own? Does it speak of some
underlying truth as yet unglimpsed? Or is it just making too much of the most
obvious trait and not looking more deeply?

On closer analysis, the most notable trait of these crystals is not their
colouration or any similarity to the elements, but their layers of symmetry of
form. Each one is comprised of a number of facets (and its strength appears to
depend on the number of facets it possesses). Each facet is identical in shape
and size to all others; furthermore, considered as a two dimensional shape, each
facet is itself comprised of identical sides and angles. It is this property
which proves most promising in understanding these structures, and which will be
analyzed henceforth.
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[1] - Incabulos Oubliette. "Continuum: Metaphysics", Library of
Universal Knowledge, Hallifax, 262 CE, page 3.
[2] - Nejii Talnara. "Studies on Harmony", Arboreal Library, Serenwilde,
256 CE, page 1.
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Geometrical Analysis
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As previously noted, the two most evident geometrical properties of the
polyhedral shapes are these:

1: Each of the facets is identical in shape and size to all other facets.

2: The shape of each facet is a perfect polygon: its sides and angles are all
congruent.

In this way, it seems that the structure of the polyhedron is perfect in its
symmetries, as it consists of identical shapes made in turn of identical shapes.
The question of why only five such shapes are evident on Continuum will be
discussed in the next section of this work.

It is evident that there are many symmetries in these shapes. Each facet is a
radially symmetrical shape, and the three-dimensional structure is also
symmetrical about all axes. The fact that the three-dimensional structure, its
formation of identical two-dimensional elements at identical angles, reflects
how the two-dimensional structure in turn is formed of identical one-dimensional
elements at identical angles, is another symmetry of structure.

There is a curious additional symmetry that is less evident but equally
important. Consider the hexahedron as a collection of vertices and edges. Its
six faces comprise twelve edges and eight vertices. Now consider the octahedron
similarly; its eight faces comprise twelve edges and six vertices. The apparent
symmetry reflects a deeper one. Take an octahedron and draw a point in the
center of each face, then connect these points, and you will find you have
traced a hexahedron. Do the same with a hexahedron and you will produce an
octahedron.

The same relationship exists between the dodecahedron (twelve faces, thirty
edges, twenty vertices) and icosahedron (twenty faces, thirty edges, twelve
vertices). Even more unusually, the tetrahedron bears the same relationship
with itself: with four faces, four vertices, and four edges, the shape created
by connecting the centers of its faces is another smaller version of itself
(interestingly, one fourth the size).

It is with this understanding of the geometric structure of the polyhedral
shapes that one can speculate on the nature and origin of the polyhedral
denizens, but first, one must answer the question why these five shapes in
particular are the only ones seen on Continuum. This will require an
application of the principles of Aesthetic Apprehension.
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Structural Principles
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It will seem at first that the five shapes seen on Continuum represent merely a
particular selection of shapes that obey the symmetries discussed previously.
However, consideration of the principles of Aesthetic Apprehension[3] and
application of mathematical rigour will demonstrate that these five shapes are
unique, and the only shapes present because no other shapes are possible.

Consider, first, these applications of the principle of Aesthetic Apprehension.
For convenience in citing them, I have relabelled them with the letters AA
before their numbers; otherwise, they correspond exactly to how they are
presented in "Continuum: Metaphysics."

AA1. A straight line segment can be drawn joining any two points.
AA2. Any straight line segment can be extended indefinitely in a
straight line.
AA3. Given any straight line segment, a circle can be drawn having the
segment as radius and one endpoint as center.
AA4. All right angles are congruent.
AA5. If two lines are drawn which intersect a third in such a way that
the sum of the inner angles on one side is less than two right
angles, then the two lines inevitably must intersect each other
on that side if extended far enough.

Starting with these as first premises, and the definition being used (that each
facet is identical to all others, and consists of a polygon of equal sides and
angles), one can prove that the five polyhedral shapes of the Continuum are the
only possible shapes, as follows. We will first have to prove a few other
assertions or "llemas" along the way, which I will label L1 through L3.
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L1: Alternate interior angles are congruent.

1. Consider parallel lines with the following angles:

/
______/_C__
D/B
/
___/A______
/
/

2. According to principles AA1, AA4, and AA5, angles B and C must add to two
right angles, and the same is true of B and D.

3. Since B and C add to two right angles, and B and D add to two right angles, C
must be equal to D.

4. Angles A and C must be equal, according to principle AA5.

5. Therefore, angles A and D are equal.
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L2: The sum of the angles in a triangle equals two right angles.

1. Consider a triangle with vertices labeled A, B, and C. According to
principles AA2 and AA5, we can draw a line through point C that is parallel with
the line that connects points A and B, and extend the line segment between A and
B.

_______C______
/\\
/ \\
____/____\\____
A B

2. The internal angles of the triangle at A and B are congruent to the angles
outside the triangle between C and this parallel line, according to llema L1.

3. The sum of the three angles at C must equal two right angles, according to
principles AA2 and AA5.

4. Since the angles at A and B equal the exterior angles at C, those angles plus
the internal angle at C must add to two right angles.
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L3: The sum of the angles in a polygon of N sides for N > 3, where all sides and
angles are congruent, is equal to 2(N-2) right angles.

1. A square, or any four-sided polygon, can be converted into two triangles by
adding a bisecting line as follows, in accordance with principle AA1:
____
|\\ |
| \\ |
| \\ |
|___\\|

2. A polygon of five sides can be divided into a four-sided polygon and a
triangle in the same way.

3. Therefore, a polygon of any number of sides N > 3 can be turned into N-2
triangles, by repeatedly adding a bisector to split it into a triangle and a
polygon of one fewer sides, until you get to a four-sided polygon, which can
then be split into two more triangles.

4. The sum of all the angles in each triangle is equal to two right angles,
according to L2.

5. Therefore, the sum of all the angles in the polygon is equal to N-2 times two
right angles.
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Proof:

1. Each vertex of a solid must be the intersection of not fewer than three
faces. This arises from the definition of a three-dimensional solid, which in
turn derives from principles AA1 and AA5.

2. The angles on each of the three or more faces that meet at each vertex must
sum to less than a full circle. If they summed to a full circle, the vertex
would have to be at a flat point, not a vertex, as implied by principle AA3, and
more than a full circle would be impossible by the same principle.

3. Since each facet must be a polygon with equal sides and angles, the angles
must therefore be less than a third of a full circle. This follows from fact
that less than a full circle is available (step 2), and it must be divided up at
least three ways (step 1).

4. The angles in such a polygon with six or more sides must be at least a third
of a circle or more, according to llema L3. Therefore, the only polygons that
are possible sides of a solid of this sort have five or fewer sides, and
therefore, can only be triangles, squares, or pentagons.

5. For triangular facets, the angle is one sixth of a circle, so a shape may
have three, four, or five triangles meeting at a vertex; any more would comprise
more than a full circle, which is impossible according to step 3. This is the
tetrahedron, octahedron, and icosahedron.

6. For square facets, the angle is a right angle, so the only possible
arrangement is three facets at a vertex, or the hexahedron, by the same
reasoning.

7. For pentagonal facets, the angle is three tenths of a circle, so the only
possible arrangement is three facets at a vertex, or the dodecahedron, by the
same reasoning.

Therefore, there are only five possible solids made of congruent facets which
are themselves formed of congruent lines and angles, and these are the five we
find on the Continuum.
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[3] - Incabulos Oubliette. "Continuum: Metaphysics", Library of
Universal Knowledge, Hallifax, 262 CE, page 8.
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Harmonic Resonance
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It cannot be coincidence that there are only five possible shapes that have
these properties, and all five of those shapes are found on Continuum natively,
and no other shapes are found there (apart from the spheres of the Master
Crystals). We must then consider what else we know of the crystals of the
Continuum.

As we learn from studying the science of Harmonics, crystals have natural
vibrational frequencies on which they resonate. A resonance is a vibration
which is attuned to the natural properties of the crystal, and thus,
self-reinforcing and self-sustaining. Imbuing a crystal, or anything else, with
a vibration at its resonant frequency will cause an ever-increasing vibration
which will eventually tear the crystal or object apart, unless it has some
property allowing it to absorb the increasing vibration indefinitely.

It is for this reason that some crystalline formations occur and others do not.
Any crystalline formation whose resonant frequency or frequencies include
vibrations to which it will be regularly exposed will tend not to form since it
will be torn apart before it can form. The only crystals that can form in a
particular environment are those which can survive the process of formation
without being resonated into destruction, either by not resonating on the
frequencies that are present, or by having a structure which can absorb and
sustain even the self-reinforcing resonant frequencies without being destroyed
by them.

Since it's known that the resonant and vibrational properties of crystals can
depend on, amongst other things, their structure, it follows that the vibrations
native to Continuum must favor the formation only of crystals that can withstand
those vibrations, and these fully-symmetrical crystals must be the only ones
which can form and survive formation in such an environment. In other words,
it's feasible that crystals of countless other structures may have a tendency to
form on the Continuum, or might have been previously formed by Dynara; but all
except for those possessing this fully-symmetrical property would have been
destroyed, perhaps going into the formation of the landscape itself.

From this we cannot infer whether these particular crystals were formed
specifically by Dynara's design, or if they were merely the only ones of the
many crystals which arose either from Her hand or from a consequence of
processes that resulted from Her creation; however, the latter seems simpler and
thus more likely. In either case, it can be concluded that the shapes of the
crystals, in accordance with the principle the Aeromancers call Existentialism,
does not follow from their purpose so much as precede it; if they have a
purpose, they suit it only because anything that could not survive long enough
to suit it wouldn't be there for us to consider.

In seeking to discover the purpose of these crystals and the role they play, we
may have learned of their origins, but more importantly, we have learned
something of the internal mechanisms by which they are formed. In time, we
might be able to study why these particular shapes have the resonant vibrational
properties they do, learn how other existing structures already behave as they
do due to similar structural formations, and employ similar shapes in new
technologies to powerful effect.
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