.context @V4000
.context @V4001
.context geom
:nGeometry
.freeze 1
\iGeom\i \b\aGeometric Doodles\b\v@L8030\v
\bHyperbolic Geometry\b
<\aThe Extentions\v@V4011\v>\p
<\aThe Extentions Revisited\v@V4012\v>\p
<\aUniform Hyperbolic Tiles\v@V4013\v>\p
<\aA More Complete Look\v@V4014\v>\p
<\aUniform Tiles in hyperbolic space\v@V4015\v>\p
<\aOctagons and Octagonny\v@V4016\v>\p
<\aThe Other extention\v@V4017\v>\p
\bThe Uniform Polyhedra and notations\b
<\aUniform Polytopes in 4D\v@V4021\v>\p
<\aNew Numbers for the Uniform Polytopes\v@V4022\v>\p
<\aNotes on old and new notations\v@V4023\v>\p
<\aThe special function\v@V4024\v>\p
<\aReflections on the order\v@V4025\v>\p
<\aNames for the figures\v@V4026\v>\p
<\aNames for the figures\v@V4027\v>\p
<\aSymbols for the view of figures\v@V4028\v>\p
\bExotica\b
<\aRegular Maps\v@V4031\v>\p
.context @V4011
:nThe Extentions
.freeze 1
\i\i \b\aThe Extentions\b\v@V4001\v
The Extentions
--------------
The first and second extension consist of those hyperbolic groups whose
every node is either at most elliptic and parabolic. These are those of
finite extent and content. In the classifications, the symmetry groups
are designated as follows: number-of-nodes form number-system. Form is
either Eliptic, Parabolic, finite Distance and Content. Number system is
1 for class 1, n (meaning span of 1, sqrt(n)) for class 2, and Xn (meaning
span of square roots of divisors for n, for class 4.
The search starts with the shape of the Dynkins-symbol, as follows: line,
branched chain, marked loop, branched loop and 7+ nodes.
Line
----
These are the regular figures:
FSQ SFS FSF FSSS FSSQ FSSF
QQS QQQ QSQS QSSQS SQSSS SSQSS
HSS HSH HSQ HSF SHS
The number systems they belong to are
4D5 FSQ SFS FSF
4C1 QQS QQQ HSS HSQ HSH SHS
4C5 HSF
5D5 FSSS FSSQ FSSF
5C1 QSQS
6C1 QSSQS SQSSS SSQSS
Trees
-----
These have three or more branches, which may be marked. For branches of
one member, removing any of the branches must leave an elliptic or
parabolic member.
QQAq QQA FSA HSA ESSQ ESSAB
Where there is a branch of 2, we add something to SSA, SQA, QSA or SSAB.
FSSA SSQA QSQA SQSA SSSAB QSSAB
For a branch of three, we need to add something to the end of SSSA or QSSA.
SQSSA.
For two branches of 2, we need to add a branch of -B to any chain of 4,
this means SSSSB, QSSSB, SQSSB or QSSQB. Of these QSSSB and QSSQB pass.
QSSSB QSSQB
Loops:
These are cycles of E or P lines. We note that S and Q are the only ones
to occur medially, and therefore for rings of 5 or 6 members, these are
always of S and Q. In rings of 4, we seek any cycle of regular figures.
SSSQ: SSSF: SQSQ: SFSQ: SFSF:
SHSS: SHSQ: SHSF: SHSH:
QQSS: QQQS: QQQQ:
SSSQS: SQSQS: SSSSSQ: SSQSSQ:
Branched loops.
We add one node to a loop, and connect it with one or more nodes on the
loop. The loop is always unmarked. When there are more than four nodes,
the non-loop node is connected in one way.
SSS:S SSS:Q SSS:F SSS:H SSSS:B: SSSS:B:B::
SSSS:S SSSS:Q SSSS:BS:
SSSSS:S
Seven-plus nodes.
While the same can be done for seven-plus nodes, there are not a lot of E
and P forms to pick from, and they can all be found by adding a node to
the special end of a P form, until some other node is also parabolic.
:t SSSSSS:S SSSSSSS:S SSSSSSSS:S
:q ESSSSB ESSSSSB ESSSSSSB ESSSSSSSB
:qr QSSSSB QSSSSSB QSSSSSSB QSSSSSSSB
:y SSSSSBS SSSSSSSC SSSSSSSSB
What we found:
D5 C1 Others
Lines 17 6 10 1
Trees 15 2 13 -
Loops 16 2 5 9
Br Loops 10 9 1
Seven+ 14 14
-- -- -- --
72 10 51 11
Here are the 11 others by number system. Apart from SSS:F which is a
subgroup of order 2 of HSF, all of these are isolated.
4D- 4C- 5D- 6C-
-1 SQSQ:
-2 SSSQ: SQQQ: SSSSQ: SSSSSQ:
-3 SSSH:
-5 SSS:F HSF
-6 HSQS:
-X10 FSQS:
-X15 FSHS:
Here are the 10 D5's allocated by dimension. FSA is a subgroup of FSQ,
and FSSA is a subgroup of FSSQ.
4D5 FSQ SFS FSF FSA SSSF: SFSF:
5D5 FSSS FSSQ FSSF FSSA
The 51 cases of C1 are now rearranged by dimension as follows. The types
are Regular (line), Tree, Loop, Branched loop and Seven+.
R T L B S :t :q :y
4C1 6 3 3 5 - 17 10 7
5C1 1 4 1 3 - 9 * 9
6C1 3 6 1 1 - 11 1 10
7C1 - - - - 3 3 1 2
8C1 - - - - 4 4 1 2 1
9C1 - - - - 4 4 1 2 1
10C1 - - - - 3 3 2 1
--------------------- -------------
10 13 5 9 14 51 14 34 3
By applying the laws of symmetry, these can be reduced to fewer rows. Here
it is done for 4-6 dimensions. In 7+, the group :q is a subgroup of order
2 of :qr.
4C1 QQS QQA QQQ QQSS: QQAq QQQQ:
HSS SSS:S SHS HSH SSS:H SSSS:B:B::
HSQ HSA SSS:Q SSSS:B:
HSHS:
5C1 SQSQ SQSA QSQA ESQA SSSS:Q SSSS:BS:
SSQSQ:
QSSB SSSS:S
6C1 SQSSS QSSSB SSQSS ESSSB SQSSQ SQSSA QSSQB QSSAB ESSAB
SQSSQS:
SSSSS:S
The groups HSHS:, SSQSQ: and SSQSSQ: can be connected to the previous
group in the table above, by noting that H/S/HS: = /HSQ, /SQSQS: = /SQSQ
and /SQSSQS: is /SQSSQ. The conjunction of the other groups requires
subtler thought. Since there is no figure that can be presented in HSS and
HSQ, or in SQSQ and QSSB, we need to find figures that share enough
elements that we can do a small count of symmetries. The groups are 8, 8
and 16 of HSQ, QSQS and QSSQS.
The figures to compare are /SHA against /SSH, and /SSQA against /EQSQ. In
both cases, the second is a subgroup of order 5 of the first.
In the first case, we note that /SHA has trilats and tetrahedra, while
/SSH has tetrahedra only. We can replace the trilats with a flat pyramid
centered on the centre of the trilat, this replaces the trilat with
tetrahedra, and the /SHA becomes an /SSH. Now, the /SSH has all of the
original tetrahedral cells, and a new one for each of the triangular faces
of the tetrahedra: all together five times. Hence SHA has regions five
times the size of SSH.
Likewise, /SSQA has an oct-tet vertix figure, and /EQSQ has oct-cuboct
vertix figure. The CO being 8 tetrahedra and 6 half-octahedra. The
octahedra of the /EQSQ coinsides with those of /SSQA, and thus we can use
this to compare the relative sizes. Since the tetrahedral margins of
/SSQS alternate from the 5-cell to the 16-cell, and of /EQSQ from the
16-cell to the oct-tets, the number of cells are then proportional to the
tetrahedral margins. Now the consider the 5-cells of /SSQS. These appear
as tetrahedral pyramids, the apex at the centre of the oct-tet, and the
base as one of the faces. That is, only one of the five faces of /SSQA
appear in /EQSQ.
4C1 :qrr QQS QQA QQQ QQSS: QQAq QQQQ:
:t HSS SSS:S SHS HSH SSS:H SSSS:B:B::
:tt HSQ HSA SSS:Q SSSS:B: HSHS:
5C1 :qrr SQSQ SQSA QSQA ESQA SSSS:Q SSSS:BS: SSQSQ: QSSB SSSS:S
6C1 :qq SQSSS QSSSB SSQSS ESSSB SQSSQ SQSSA QSSQB QSSAB ESSAB SQSSQS:
:t SSSSS:S
7C1 :qr QSSSSB ESSSSB
:t SSSSSS:S
8C1 :y SSSSSBS
:qr QSSSSSB ESSSSSB
:t SSSSSS:S
9C1 :y SSSSSSSC SSSSSSSS:S
:qr QSSSSSSB ESSSSSSSC
10C1 :y SSSSSSSSB QSSSSSSSSB ESSSSSSSSB
The enumeration of the separate symmetry groups follow,
QQS 1 48 12 8 8
QQA 2 96 96 8 12
QQQ 3 U 48 48 U 48 /SQQ
QQSS: 4 8 96 96 \\QQA Q\\S is double-/SQ
QQAq 6 U 48 U U U
QQQQ: 12 U U U U
SSH 2 4 48 4
SSS:S 4 4 20
HSQ 5 200 20 UU 48
SHS 8 48 48 200
HSA 10 200 20 U
SSS:Q 10 20 UU UU
HSH 12 U 48 U
SSSS:B: 20 20
SSS:H 24 48 U
SHSH: 40 200 200
SSSS:B:B:: 48 48
QSSA 3 300 6 6
QSQS 5 4800 60 200 60
SSSS:S 6 6 200
SQSA 10 200 60 300
QSQB 15 4800 60 U 800
QSAB 30 4800 60 4800
SSSS:Q 30 60 U U
SSQSQ: 40 800 200
SSSS:BS: 60 60
SQSSS 1 6 100 6
QSSSB 3 6 4800 100
SSQSS 5 100 100 3200
ESSSB 6 6 4800
SQSSQ 10 3200 100 U U
SQSSA 20 3200 100
QSSQB 30 U 100 U
QSSAB 60 U 100
ESSAB 100 100
SSQSSQ: 140 3200 3200
SSSSS:S 1 1 Unrelated to SQSSS
The groups D5
The numbers for 5D5 come from the Euler-excess for 2.0000 verties. By
laws of symmetry we know FSA = 2 FSQ, FSSA = 2 FSSQ; and
VFSS/4=FVFS/6=FSSS.
FSQ 1
SFS
FSF
FSA 2
FSSS:
FSFS:
FSSS 2
VFSS 8
FVFS 12
FSSQ 17
FSSA 34
FSSF 52
Euler-excess says that for a sphere, the excess of the even-dimensions
less the odd-dimensions is proportional to the area of the sphere. We
use this to evaluate the size of the 5D5 groups.
N0 N1 N2 N3 N4 Excess Per 2.0000 zones.
FSQ 500 1500 1200 200 0
SFS 200 2000 2000 200 0
FSF 200 1200 1200 200 0
FSSS 200 500 400 100 2 2
FSSQ 75 260 300 100 2 17
FSSF 2 100 248 100 2 52
.context @V4012
:nThe Extentions Revisited
.freeze 1
\i\i \b\aThe Extentions\b\v@V4001\v
There are only a limited number of hyperbolic groups of finite content
and simplex regions, past three dimensions. The following is an search
for those 72 groups in 4 or more dimensions, that yield elliptic or
parabolic groups on the removal of any node.
The search procedes according to the topology of the Dynkins symbol.
17 Chain This is a line of branches, always regular
26 Trees This refers to groups with branching points
13 Rings This is an unmarked ring with a branch.
16 Loops These are marked rings.
These are then sorted according to how many nodes they have and the
number system.
\bChains\b
4D5 FSQ SFS FSQ
4C1 HSS HSQ HSH SHS QQS QQQ
4C5 HSF
5D5 FSSS FSSQ FSSF
5C1 QSQS
6C1 SQSSS SQSSQ SSQSS
\bTrees\b
This is by far the most difficult to find. All of these are C1, except the
FSA and FSSA, which are D5.
4D HSA QQA QQAq FSA 1,1,1
5D QSAB 1,1,1,1
SSQA SQSA QSQA FSSA 2,1,1
6D ESSAB 1,1,1,1,1
SSSAB QSSAB 2,1,1,1
QSSSB QSSQB 2,2,1
SQSSA 3,1,1
7D ESSSB QSSSSB
8D ESSSSB QSSSSSB SSSSSBS
9D ESSSSSB QSSSSSSB SSSSSSSC
10D ESSSSSSB QSSSSSSSB SSSSSSSSB
\bRings\b
Since an unmarked ring is parabolic, there is only one node off it. If
this node is connected to the ring more than once, no more than one node
can be on each limb between the trivia.
All of these are C1, except the C5 SSS:F
4D SSS:S SSS:Q SSS:F SSS:H SSSS:B: SSS:AS:A::
5D SSSS:S SSSS:Q SSSS:BS:
6D SSSSS:S
7D SSSSSS:S
8D SSSSSSS:S
9D SSSSSSSS:S
\bLoops\b
This is a loop of regular figures. A ring of 4 may yield any regular
figure, but rings of 5 and 6 can only be comprised of S and Q, since these
are the only two that occur medially.
The number system is augmented by the loop factor, which is evaluated as
the product of all of the bridge constants around the loop. If this is
not in the number system, then the span of the number system is increased
by this factor.
The affected braches are Q (sqrt 2) and H (sqrt 3).
4C1 QQSS: QQQQ: HSHS: 4D1 QSQS: 5C1 SQSQS: 6C1 SSQSSQ:
4C2 QQQS: 4D2 QSSS: 5D2 SQSSS: 6C2 SSQSSS:
4C3 HSSS:
4D5 SFSF: SSSF:
4C6 HSQS:
4DX10 QSFS:
4CX15 HSFS:
\bClassification by number\b
In the breakup by number system, the 4 digits refer to how many C T R and L
of each number group. Thus 6353 means 6 L, 3 T, 5 R and 3 L items.
Total by type Breakup by Nr system
C T R L C1 C1 D5 C5 Oth
4 10 4 6 12 32 17 6353 31-2 1-1- ---7
5 4 5 3 2 14 9 1431 31-- ---- ---1
6 3 6 1 2 12 11 3611 ---- ---- ---1
7 2 1 3 3 -21- ---- ---- ----
8 3 1 4 4 -31- ---- ---- ----
9 3 1 4 4 -31- ---- ---- ----
10 3 3 3 -3-- ---- ---- ----
17 26 13 16 72 51 10 2 9
The fifty-one C1 figures can be reduced by the operation of the laws of
symmetry and identity of polytopes as follows. The identities used are
as follows:
/KQ..QK: = /KQ..Q (for all K); .. is any number of S
o2PxPx2PoKz = x2Po3oKKo K,KK as 2,3; 3,4 ( since s2sKs = x3oKKo )
4:qq QQS QQA QQQ QQSS: QQAq QQQQ: 6
4:tt HSS SSS:S SHS HSH SSS:H SSSS:AS:A:: 6
4:t HSQ SSS:Q HSA SSSS:B: HSHS: 5
5:q SSQA SSSS:S 2
5:qrr QSQS SQSA QSQA QSAB SSSS:Q SSSS:BS: SQSSQ: 7
6:qq SQSSS QSSSB SSQSS SSSAB
QSSQS ESSQS QSSQB QSSAB ESSAB SSQSSQ: 10
6:t SSSSS:S 1
7:t SSSSSS:S 1
7:qr QSSSSB ESSSSB 2
8:t SSSSSSS:S 1
8:qr QSSSSSB ESSSSSB 2
8:y SSSSSBS 1
9:t SSSSSSSS:S 1
9:qr QSSSSSSB ESSSSSSB 2
9:y SSSSSSSC 1
10:qr QSSSSSSSB ESSSSSSSB 2
10:y SSSSSSSSB 1
One can compare partial segments to illistrate that there are further
subgroups. Specifically, 4:t 4:tt, and 5:q 5:qrr are related, and
9:y 9:t, and 10:qr 10:y may be related. The other codimensional pairs such
as 4:qq to 4:tt, 6:t to 6:qq, any of the 7D or 8D groups, or 9:qr to 9:y or
9:t may be entirely unrelated.
The 10 D5 and 2 C5
The laws of symmetry makes the second in each row, a subgroup of the first.
The rest have the same number system, but do not equate to either.
4D5 FSQ FSA SFS FSF FSA SSSF: SFSF:
4C5 FSH SSS:F
5D5 FSSQ FSSA FSSS FSSF
The vertix figure of /K/LMN: is an M antiprism, the sides of which are
the shortchords of L and N, and the lacing is the shortchord of 2K. Thus
/S/HSH: is a triangular antiprism, top, bottom and lacing are all H, this
makes it the same as /HSQ.
p top bottom lacing
/S/HSH: 3 6 6 6 /HSQ
/S/SFS: 5 3 3 6
/S/FSS: 3 5 3 6
/S/FSF: 3 5 5 6
/F/SSS: 3 3 3 10
/F/SFS: 5 3 3 10
/bLoops/b
The nine other number systems contain only one element. There are no
subgroups. The four with octagonal symmetry (s=2) may be regarded as
sections of the final 6C2 SSSSSQ: As a result, we may be able to compare
SSSQ: and QQQS:.
s 4Ds 4Cs 5Ds 6Cs
1 SQSQ:
2 SSSQ: QQQS: SSSSQ: SSSSSQ:
3 HSSS:
6 HSQS:
10 QSFS:
12 HSFS:
.context @V4013
:nHyperbolic Tiles
.freeze 1
\i\i \b\aHyperbolic Tiles\b\v@V4001\v
The following polyhedra tesselate hyperbolic space. The requirement is
that all edges must be equal. There are an infinite number of cases of
the type "rxPxQoRo", as long as PQ is elliptic and QR hyperbolic. The
faces of the xPxQo fall into two hyperbolic tilings x2Po4o and xQoRo.
Since there are five values of PQ, we can find the matching R, the only
case that yields integer R is 4-3-8.
The cells and vertix figures are given as Dynkins symbols for 4D and
Wythoff symbols for three dimensions. Where it is not given, an irregular
tetrahedron is assumed. The vertix array is the vertix figure expanded to
show the cells instead of its faces. The sections refer to planes that
contain these figures. Thus, the octagons of the truncated cubes fall in
an {8,4}, and the triangles fall in {3,8}. Not all of these honeycombs
have sections like this.
cell vf vertix array Sections
x5o3o4o 3 | 2 5 4 | 3 2 8 dodecah {5,4}
o5o3o4x 3 | 2 4 5 | 3 2 20 cubes
x3o5o3o 5 | 3 2 3 | 5 2 12 icosah
x5o3o5o 3 | 5 2 5 | 3 2 20 dodecah
o3x5x3o 3 2 | 5 | 2 2 2 4 tr dodecah
o5x3x5o 5 2 | 3 | 2 2 2 4 tr icosah
x4x3x4x3z 2 3 4 | | 2 2 2 4 tr cuboctah
x5x3x5x3z 2 3 5 | | 2 2 2 4 tr icosadodah
rx4x3o8o 2 3 | 4 8 2 || 2 16 tr cubes {3,8}, {8,4}
4: Tetrahedra. These have opposite edges equal.
8: Octahedron. {3,4}
12: Dodecahedron {5,3}
16: Octagon dipyramid
20: Icosahedron {3,5}
In four dimensions, we find
cell vf vertix array Sections
x5o3o3o3o x5o3o3o x3o3o3o 5 twelftycells
o5o3o3o3x x3o3o3o x3o3o5o 500 5-cells {3,10}, {3,5,A}
x5o3o3o4o x5o3o3o x3o3o4o 16 twelftycells {5, 4}, {5,3,4}
o5o3o3o4x x4o3o3o x5o3o3o 500 tesseracts {4,10}, {4,5,A}
x5o3o3o5o x5o3o3o x3o3o5o 500 twelftycells {5,10}, {5,5,A}
o5o3x3o5o o3x3o5o o5x2x5o 10 2tr.fifhundcells octa diam {4,5}
o3rx4x3o8o o3x4x3o o8m2m8o 64 octagonny {3,8} {8,4}, rx4x3o8o
o3rm4m3o8o o8x2x8o o3m4m3o 248 bioctagons {8,6} {4,8}
5: 5-cell {3,3,3}
10: pentagon-pentagon prism
16: 16-cell {3,3,4}
64: octagon-octagon cross
248: dual of octagonny: convex hull of dual {3,4,3}'s
500: fifhundcell {3,3,5}
Laminahedra.
------------
The term laminahedra refer to a polyhedra bounded by unbounded sides. The
class leader are slabs bounded by two parallel planes, or strips bounded
by two unbounded lines. Because the sides is unbounded, laminahedra of
any shape and size (in the same dimension) tile space.
The most interesting cases of laminahedra are the inscribed laminahedra.
Here the sides are inscribed with a honeycomb of cells, and the content
expanded out to its components. If we tile space with these laminahedra,
then we can also tile it with its inscriptions. An example of an
inscribed laminahedra is the aperigonal antiprism, which is a strip of
triangles. Another is the aperigonal prism, a strip of squares. One can
tile 2-space with these strips of triangles or squares, or alternate
between them.
Before we look at hyperbolic cases, we will consider the rx2x3o6o. This
is a laminate reflection of the x2x3o6o, the r means that the node that it
falls on is a flat plane, and therefore is a mirror. The x2x3o6o can be
inscribed in a laminahedra, which we reflect with r.
The x2o3o is a hosohedra, a line-segment with three sides. On a sphere,
it would appear as three lines of longitude, from N to S pole, 40 apart.
If the line is infinite, we could expand the sides to non-zero with, the
result being an infinitely long triangular prism. The x2x3o is a
truncation of this, set so that all sides are equal. we simply cut off a
peice of our infinitely long prism, so that the length and height are
equal.
Now, the x2o3o6o is a tiling of line segments, which when made infinite,
we resolve into infinite triangular prisms of finite section. The
truncate of this is x2x3o6o, which has triangular prisms x2x3o and two
sides x3o6o. Since the x3o6o is flat, we can replace it with a mirror to
reflect the previous side, the mirror falls at the node marked r in
rx2x3o6o.
The triangular prism has two types of faces: exposed triangles, and buried
squares (with respect to the laminahedra), or exposed squares and buried
triangles in respect to the infinite prisms (which are called yicles
"spears"). Yicles are bounded by laminahedra: their margins are unbounded,
and their faces are bounded by laminahedra.
We now head to hyperbolic space. It is not possible to have two planes
that are parallel to each other, but we can still construct laminahedra.
Consider the figure oPoQoRx, where PQ is elliptic and QR is parabolic.
This figure is bounded by unbounded faces oQoRx. The faces are convex,
and therefore there lies a flat face that is completely equidistant from
the verticies of the convex face. In order to use this face as a mirror,
we must construct a figure that has some form of QR as a face, and other
faces, the example used is xPxQoRo. This has xPxQo and xQoRo as faces,
and two degrees of freedom (2 x's) However the degrees of freedom do not
specifically imply that when the edges are equal the xQoRo is flat.
If we now take an xPxQoRo with a flat xQoRo, we can us the xQoRo as a
mirror and have space tiled entirely with xPxQo. But the edges of xPx are
not always equal. The faces of xPxQo are xQo and xPx. The xQo are
always regulat and are exposed in the laminahedra xQoRo. The xPx are
2p-gons these are exposed in the yicles. We note that the angles of the
2p-gons are right angles (30), and thus if the edges are equal, they are
the same size as those in x2Po4o. We need only look for cases where PQ is
elliptic and xQoRo and x2Po4o are the same edge-lengths.
2,Q x4o4o = xQoRo rx2x3o6o rx2x4o4o rx2x6o3o
3,3 x6o4o = x3oRo
3,4 x6o4o = x4oRo
3,5 x6o4o = x5oRo
4,3 x8o4o = x3oRo rx4x3o8o
5,3 x10o4o = x3oRo
4,4 x8o4o = x4oRo
3,6 x6o4o = x6oRo rx3x6o4o
6,3 x12o4o = x6oRo
rx2x3o6o Triangular prisms
rx2x4o4o Square prisms, this is x4o3o4o
rx2x6o3o Hexagonal prims
rx4x3o8o Truncated cubes
rx3x6o4o Truncated trilats this is x6o3o4o
In four dimensions, these can be extended by the ruse of
o3rx2x3o6o Bitriangles
o3rx4x3o8o Octagonny
In this case muliple reflectios are required. The base form o3x4x3o8o has
two faces o3x4x3o and x4x3o8o. While we need mirrors to replace the
x4x3o8o's, the required mirrors lie at the faces of the x4x3o, not the
laminate plane. To this end, we put the laminate reflections at the node
opposite the x3o8o.
.context @V4014
:nA more complete Section
.freeze 1
\i\i \b\a4D Hyperbolic groups\b\v@V4001\v
This is the latest on oPo3o6o
:tt :tq :tf
1 5 HSQ uab
2 10 SSS:Q ua
2 HSS 10 HSA ab
4 SSS:S 20 SSSS:B: a
8 SHS 40 HSHS: bb
12 HSH ubu
24 SSS:H bu
48 SSSS:B:B:: 200 b
The group of order 200 is the cubic cells of /QSH. Since the FSH contains
a group of order 100.FSH, these dodecahedra, we note that the group marked
b is a cell of xPo3o6o, and it holds any subgroups that one might apply to
the tetrahedron, cube or dodecahedron.
The only symmetries of the tetrahedron are bilateral, digonal and
triangular pyramid, apart from its complete symmetry. If we admit these
to the symmetry of the tetrahedron, we get the following groups. Note
except for the complete symmetry, removing the tetrahedral walls makes the
groups infinite.
t c Size Result c Size Result
o3o3o y 2 SSH n 4 SSS:S
o3o y 8 SHS n SSSS:B:B::
o2o y 12 HSH n
o y 24 SSS:H n
- y 48 SSSS:B:B:: n
For the cube, there are two kinds of symmetry, the prismatic r and the
group q.
r q c Size result c Size
o2o2o o3o3o y 5 HSQ n 10 SSS:Q
o3o3o y 10 HSA n 20 SSSS:B:
o2o2o o2o y 15
o2o2o y 30 (cube 1) n cell of x6o3o4o
o2o o2o y 30
o o2o y 30
o3o y 40 SHSH: n cell of o6o3o4x
o2o y 60 (cube 2) n infinite
o2o y 60 (tri qt)
o o y 60
o y 100 (cube 4) n infinite
o y 100 (tri h) n infinite
y 200 x4o3o6o
The (cube n) is n octants where 8 make a cell of x4o3o6o. These have n
points at infinity. The tri qt and h is a sq prism cut on the diags of
the square.
The dodecahedron yields the following. While it has a lot of elements, it
does not have a lot of subsymmetry subgroups.
f c Size Result c
o3o5o y 1 FSH n SSS:F dodecahedral
o5o y 12 n pentagonal slice
o2o2o y 15 n three right planes
o3o y 20 triangular slice
o2o y 30 two mirrors at an edge
o y 60 just one mirror
y 100 no internal mirrors.
The o4o4o3o.
------------
There are six simplex members of this marked X in the list. This group
splits into lots of degenerate groups, as does QQ. Nodes marked a belong
to a group of size 8. Nodes marked u belong to an infinite-sized group
(much like the strips of QQ.), those marked o are finite groups that
elsewhere degenerate, and nodes marked - merge adjacent symmetry regions.
a
X o4o4a3a 1 8 48 12 abc
X o4a4oAa 2 8 96 96 add
X u4o4o4u 3 48 48 ccuu
3 48 buu This appears to be equal to ccuu.
X o4a3a3o4z 4 8 96 ad
X u4o4uA4u 6 48 cuuu
6 bu
8 8 a
X u4u4u4u4z 12 - uuuu Tetrahedron = 2 octahedral faces.
12 96 96 b Triangular biyramid (all right angles)
u4u4u4-4z 24 Half-octahedron.
u4-4u4-4z 48 c Octahedron = cell of x3o4o4o
96 d Rhombic Dodecahedron o3m4o.
u4u4-4-4z 96 dual of an s2s4s.
u4-4-4-4z U u Cell of QQQ.
The group of 12 is a cell of o4m4o3o. The shape is a triangular antiprism
with all sides 30. This has the required vertix-split of three and two.
This has subgroups of 6,6,3,2,1. Just as 3q is the degenerate group equal
to rr, the group 4:q in 4:qrr is a right-angle-only groups, this one and a
group of 48 being the octahedral cells of x3o4o4o.
The group gets frighteningly complex, partly because the nature of mirrors
of 2qrr are identical except position and size.
Consisder the QQQQ: (12). This has four edges of angle 15 and two of 30.
If four of these are joined around the 30-edge, the 15-edges join in
pairs, and an right octahedron emerges (48). The octahedron can be
divided into eight octants (6), this has three edges 30 and 3 base edges
15. If 8 of these are added to the face of a right octahedron, new faces
form along the edges of the octahderon. This figure is a right rhombic
dodecahderon (96). This rhombic dodecahedron is the mirror-planes c in
c4o4cAo, c4o3o3o4z.
Going back to our QQQQ: (12) again, this has four faces. If we connect
two tetrahedra through a face, we end up with u4u4u4-4z, a
half-octahedron. To get the full octahedron, we remove two mirrors that
at 30, this yields a figure o4-4o4-4z. (48). This figure can not be a
submember of QQQQ: because it has verticies at points other than infinity.
The figure o4o4-4-4z merges the faces of QQQQ: around angles of 15. This
makes a different figure of 96 to the one described above: This one is an
octahedron with quadralateral faces, the dual of a square antiprism.
There is a zigzag equator made of eight edges of angle 15, and eight edges
to the alternate poles each of 30.
This leaves the group size 8 to be described. This group is formed from
QQS by reflecting in the two sides Q. It evaluates to the dual of "s2s4s
square antiprism". The dihedral angles are 20 on the equator and 30 on
the meridiums. The mirrors are all the same because they are connected by
odd angles (20).
Other groups
SSSQ: SFSQ: SSSH: SFSH: SSSSQ: and SSSSSQ: have just one mirror type.
SHSQ: has a subgroup of size 6, two copies provide the mirrors for this.
The shape is a cube, all angles 20 except three that connect to one of the
verticies. The group of 6 can divide into 2 of three by a single mirror.
QQQS: has two subgroups, one of 4 and the other of 48. This second one
has lots of subgroups, the overall shape being a right-angled m3o4m. The
group of 4 consists of a octahedron, all angles right except two opposite
sides in one of the hemispheres.
SQSQ: has a subgroup of size 6, this is a cube where the petrie polygon is
20 and the meridians 30. This taken twice provides the mirrors for SQSQ:.
.context @V4015
:nUniform Tiles
.freeze 1
\i\i \b\aUniform Tiles\b\v@V4001\v
The uniform tiles in hyperbolic space.
Schafli Wythoff Krieger Krieger
{3,3} 3 | 2 3 x3o3o (none)
{3,4} 4 | 2 3 x3o4o (none)
{4,3} 3 | 2 4 x4o3o x4o3o5o {4,3,5}
{3,5} 5 | 2 3 x3o5o x3o5o3o {3,5,3}
{5,3} 3 | 2 5 x5o3o x5o3o5o {5,3,5}
x5o3o4o {5,3,4}
t{3,3} 2 3 | 3 x3x3o (none) lam rx3x3o>6o
t{3,4} 2 4 | 3 x3x4o (none) lam rx3x4o>4o
t{4,3} 2 3 | 4 x4x3o rx4x3o8o lam rx4x3o>6o
t{3,5} 2 5 | 3 x3x5o o5x3x5o lam rx3x5o>3o
t{5,3} 2 3 | 5 x5x3o o3x5x3o lam rx5x3o>6o
r{3,4} 2 | 3 4 o3x4o (none) looping
r{3,5} 2 | 3 5 o3x5o (none) looping
rr{3,4} 3 4 | 2 x3o4x cube
rr{3,5} 3 5 | 2 x3o5x (none) looping
tr{3,4} 2 3 4 | x3x4x x3x4x3x4z sublooping
tr{3,5} 2 3 5 | x3x5x x3x5x3x5z sublooping
s{3,4} | 2 3 4 s3s4s dodeca
s{3,5} | 2 3 5 s3s5s dodeca
Laminates.
A laminahedron is a ployhedron bounded by unbounded sides. The
nominative member is a slab formed by two parallel planes (lamina means
layer). Clearly, laminahedra tile space. A laminahedron can be used to
carry in other things. For example, a strip can be filled with triangles
or squares, and these laminahdera stacked in any order.
In three dimensions, the laminahedra have inscribed sides. For example
a plane of hexagonal prisms has inscribed {6,3} on each face. Likewise
triangular prisms and oct-tet layers have {3,6}. We can stack any
combination of oct-tet trusses and triangular prism to form a lattice,
there are three uniform ones.
Consider the case PQR where PQ is elliptic and QR is parabolic. As long
as QR is parabolic, the edges of RQP are finite. The cells do not need
to, and it is a feature of the hyperbolic space that a lattice can have
a dihedral angle. There is one vertix of the fundemental region that is
infinite, the one marked x in xPoQoRo. There is a certian truncate
xPxQoRo, where the face xQoRo is perfectly flat. In this case, the xPxQo
are wedged in laminahedra.
If we now have xPxQoRo, the xQoRo can be made flat. This xPxQo then form
a packed laminahedra which can tile space with any other laminahedra
marked with xQoRo (because all flat xQoRo are the same size.) Thus, we
could have space tiled with laminahedra packed with x5x3o and x4x3o,
sharing a common x3o120o. To force a reflection at the required node,
the r node-actity is used to produce a laminahedral mirror. Thus rxPxQoRo.
Only some laminahedra have equal edges. The necessary condition is that
the x2Po4o and xQoRo have equal edges. This quickly resolves to the
following values of rxPxQoRo.
rx2x3o6o, rx2x4o4o, rx2x6o3o, rx4x3o8o.
Of these, the first three are layers of trianglur, square and hexagonal
prisms. The fourth one is truncated cubes, 16 at a corner.
Since the x4x3o is a face of o3x4x3o, we might ask does can 4-space be
divided into o3x4x3o by planes that cut with rx4x3o8o? Yes. The result
is something like o3rx4x3o8o, it has 64 o3x4x3o at each vertix, the vf
being o8m2m8o. The dual has bioctagon prisms, 248 at a vertix.
Looping.
In the case of o3xPo and x3oPx, there are four faces at a vertix. The
case of o3x5o is explored here, the others are simmilar by substitution.
The vertix figure of any figure that has o3x5o as cells, is bounded by
quadralaterals with sides marked 3 and 5. Look at the sides that are
connected to any given face by, say a '3' margin. Since the faces stack
into an unbranchable chain, the only possible combination is to form an
unending line, be it circle or line. Because, the 'five' sides are doing
the same thing, the only solution for this is to have an even number of
quadralaterals at each vertix. The vertix can be topologically equal to
a {4,2p}.
When p=1, the figure turns out to be two o3x5o back-to-back. When p=2,
the vertix figure is space tiled with rectangles o3v5o. These have
cusping verticies at infinity. When p=3 or greater, the verticies are
transfinite, and one can replace a plane with the topological equal of
x4o2Po as the laminate plane.
Sublooping
Thus, while the o3xPo and x3oPx may not yield uniform lattices, the
o3xPo truncates to x3xPx, which may be truncated by flat x4o2Ro. In this
way, space might be filled with x3xPx. But what topology admits, the
circle drawing rules take. None of these has a ghost of a chance, since
the x4o6o demands a vertix figure greater than what the triangles are
prepared to yield.
The cube and dodeca.
The idea of space tiled with snub faces should not be lightly dismissed.
The x3o5o3o is tiled with snub tetrahedra.
Another opportunity for x3o4x is a marked cube. This is shares with the
pentagonal-vertix figures s3sPs, which rely on marked dodecahedra.
In both cases, these figures have a vertix figure where one of the sides
is different to the others: a triangle v 3 squares, or a P-gon v 4
triangles.
The vertix figure can then becomes marking three of the cubes edges or
six of the dodecahedron's edges so that each of the faces has just one
marked face. This can be done uniquely for the cube, and in several
different ways for the dodecahedron.
For the cube, this solution marks alternate sides of the petrie polygon.
For the dodceahedron, there is more than one way to mark the sides.
The process is rather complex, and I think there needs to be some sort
of deeper thought required to dismiss the case of s3s5s and s3s4s into
the dodecahedron, or x3o4x into the cube.
This completes the enumeration for the 3D case.
.context @V4016
:nOctagons and Octagonny
.freeze 1
\i\i \b\aOctagons and Octagonny\b\v@V4001\v
\bOctagons\b
It was decided to test the notion that the octagonny of S/Q/SP8 were the
same size as those in S/Q/SSS:. The edges of #S/Q/SP8 are the same as /SP8,
since the vertix figure is a bioctagon cross, of octagons edges equal to
the radius.
\bThe vertix figure of ..SK/L/M:\b
In the following discussion, .. means zero or more units of S, eg X..Y means
XY, XSY, XSSY, XSSSY, etc. By itself, it has the same meaning. This does
not prevent X or any other letter holding a value of S.
The series of figures K/L/M:, SK/L/M:, SSK/L/M: etc all have similar
vertix figures, being an line, S, SS, ... antiprism. This is the same shape
as the cross polytope of the same dimension, but expanded or contracted in
a direction normal to one of the faces. The base and top of these are
simplexes, of lengths /k and /m, and the lacing (all other edges) are the
same as the shortchord of 2l.
Since a simplex contains all lesser-order simplexes, the verticies of a
..K/L/M: lie in all greater ..SK/L/M:, but there is no need for these to be
diametric. This happens only when K=M.
When L=2, the figure resolves to /M..K/, so what we say of ..K/L/M: is
true of /M..K/.
In the case of /S..Q/, the higher figures contain lesser ones, not in a
centric plain, but parallel to the plain containing the centre. This is
the same way that the octagon appears in /SQ/. In the case of /S..F/, the
figure changes geometry /SF/ and /SSF/ are elliptic, /SSSF/ is parabolic.
In the cases of /Q..Q/ and /S..S/, the middle section are manifestly a lesser
/Q..Q/ and /S..S/, and therefore these are sections. This is also the case
for /F..F/ and /H..H/.
In the case of ..K/L/K:, this has a section of K/L/K:, which is the same
as K/(2L). This means that the vertix figure for all ..K/L/K is the same
as K/(2L), the diameter being the sum of the shortchords of K and 2L.
K L 2L vfd
/S..S/ 3 2 4 3.000000
/Q..Q/ 4 2 4 4.000000
/S..S/S: 3 3 6 4.000000
/S..S/Q: 3 4 8 4.414213 /SSS/Q: /SSSS/Q: /SSSSS/Q: /SSSSSSS/Q:
/F..F/ 5 2 4 4.618033 /FSF/ /FSSF/
/S..S/F: 3 5 10 4.618033 /SSS/F:
/S..S/H: 3 6 12 4.732050 /SSS/H:
/H..H/ 6 2 4 5.000000 /HSH/
/Q..Q/S: 4 3 6 5.000000 /QSQ/S: /QSSQ/S:
/Q..Q/Q: 4 4 8 5.414213 /QSQ/Q:
/F..F/S: 5 3 6 5.618033 /FSF/S:
/H..H/S: 6 3 6 6.000000 /HSH/S:
The vertix figure of these are two opposite simplexes of edge K, with a
lacing of 2L. When K=2L, there is no distortion, and the thing is regular,
as /K..Q, such happens with, eg /Q..Q/ and /H..H/S:.
Of the loop groups, this leaves only QQSS:, QQQQ:, SSQSSQ:, QSHS:, FSQS:,
and FSHS:. Of these, QQSS: and QQQQ: derive from symmetries of QQS, and
SSQSSQ: from SQSSS.
.context @V4017
:nOther extent rearangement
.freeze 1
\i\i \b\aOther extent rearranged\b\v@V4001\v
The latest thinking is to group the second extent according to seven classes
based on the topology of the dynkins symbol, and for loops, the looping
constant.
C chain regular figures
T tree branched chains
R ring rings with branches
L loops loops with even Q and even H
2 loops 2 loops with odd Q and even H
3 loops 3 loops with even Q and odd H
6 loops 6 loops with odd Q and odd H
Groups are pentagonal if they have a branch F in them, Q if there are an odd
number of Qs in the loop, and H if there is an odd number of H in the loop.
This makes for Q, H, QH, F, FQ and FH.
C T R L 2 3 6 C T R L 2 3 6
3 6 - - 7
4 6 3 5 4 2 1 1 4 1 1 2 1 1 - 32
5 1 4 3 1 1 3 1 14
6 3 6 1 1 1 12
7 2 1 3
8 3 1 4
9 3 1 4
10 3 3
10 24 12 6 4 1 1 7 2 1 2 1 1
<-----52-----> 4 1 1 <---12----> 1 1 72
\iChains\i
4 QQS QQQ HSS HSQ HSH SHS FSQ SFS FSF FSH
5 QSQS FSSS FSSQ FSSF
6 QSSQS SSQSS SQSSS
\iTrees\i
4 QQA QQAq HSA FSA
5 SQSA QSSB QSQA QSAB FSSA
6 SSSAB QSSSB QSSQB SQSSA QSSAB ESSAB
7 QSSSSB ESSSSB
8 QSSSSSB ESSSSSB SSSSSBS
9 QSSSSSSB ESSSSSSB SSSSSSSC
10 QSSSSSSSB ESSSSSSSB SSSSSSSSB
\iRings\i
4 SSS:S SSS:Q SSS:H SSSS:B: SSSSB:B:: SSS:F
5 SSSS:S SSSS:Q SSSS:BS:
6 SSSSS:S
7 SSSSSS:S
8 SSSSSSS:S
9 SSSSSSSS:S
\iLoops\i
4 SQSQ: SSQQ: QQQQ: HSHS: SSFS: SFSF:
5 SQSSQ:
6 SSQSSQ:
\iLoops 2\i
4 SQSS: SQQQ: SFSQ:
5 SSQSS:
6 SSSQSS:
\iLoops 3\i
4 SSSH: SFSH:
\iLoops 6\i
4 SQSH:
.context @V4021
:nUniform Polytopes
.freeze 1
\i\i \b\aUniform Polytopes in 4D\b\v@V4001\v
Figures, according to http://members.aol.com/Polycell/
Uniform figures in 4D, made of polygons of equal edge, and with alike
verticies. Wythoff's construction by mirror-edge makes most of these.
Here, a uniform polychora has uniform polyhedral faces.
0 10 20 30 40 50 60
0 x4o3o3o x4o3x3x x3x4x3x o5x3o3x o4x3oXx x5x3oXx
1 x3o3o3o o4x3o3o x4x3x3x s3s4x3x o5o3x3x o4o3xXx x5s3xXx
2 o3x3o3o o4o3o3x x3o4o3o x5o3o3o x5x3x3o x4x3oXx o5x3xXx
3 x3x3o3o x4x3o3o o3x4o3o o5x3o3o x5x3o3x x4o3xXx x5x3xXx
4 x3o3x3o x4o3x3o x3x4o3o o5o3x3o x5o3x3x o4x3xXx s5s3sXx
5 x3o3o3x x4o3o3x x3o4x3o o5o3o3x o5x3x3x x4x3xXx
6 o3x3x3o o4x3x3o x3o4o3x x5x3o3o x5x3x3x s4s3sXx
7 x3x3x3o o4o3o3x o3x4x3o x5o3x3o j5j2j5j x5o3oXx xPoXxQo
8 x3x3o3x x4x3x3o x3x4x3o x5o3o3x x3o3oXx o5x3oXx sPs2sXo
9 x3x3x3x x4x3o3x x3x4o3x o5x3x3o x3x3oXx o5o3xXx
#56 = s4s3sXx #27 = o3x4x3o "Octagonny"
( )---( )---( ) (o) ( )---(o)---(o)---()
4 4
The symbol j5j2j5j is a pseudosymbol for a figure with 20 pentagonal
antiprism and 2.60 tetrahedra. Such can be made by removing the twenty
verticies on polar circles of x3o3o5o.
The symbol sPs2sXx denotes a prism of the p-gon antiprism
Trace Wythoff Schafli Name
1 x3o3o 3 | 3 2 {3,3} Tetrahedron
2 x4o3o 3 | 4 2 {4,3} Cube, Hexahedron
3 o4o3x 4 | 3 2 {3,4} Octahedron
4 x5o3o 3 | 5 2 {5,3} Dodecahedron
5 o5o3x 5 | 3 2 {3,5} Icosahedron
6 o3x4o 2 | 4 3 r{4,3} Cuboctahedron
7 o3x5o 2 | 5 3 r{5,3} Icosadodecahedron
8 x3x3o 2 3 | 3 t{3,3} Truncated tetrahedron
9 x4x3o 2 3 | 4 t{4,3} Truncated Cube
10 o4x3x 2 4 | 3 t{3,4} Truncated octahedron
11 x5x3o 2 3 | 5 t{5,3} Truncated dodecahedron
12 o5x3x 2 5 | 3 t{3,5} Truncated icosahedron
13 x4o3x 3 4 | 2 rr{3,4} Rhombicuboctahedron
14 x5o3x 3 5 | 2 rr{3,5} Rhombicosadodecahedron
15 x4x3x 2 3 4 | tr{3,4} Truncated Cuboctahedron
16 x5x3x 2 3 5 | tr{3,5} Truncated Icosadodecahedron
17 s4s3s | 2 3 4 sr{4,3} Snub cuboctahedron
18 s5s3s | 2 3 5 sr{5,3} Snub icosadodecahedron
A xXxPo 2 p | 2 t{2,p} p-gonal prism {p}x{}
B s2sPo | 2 2 p sr{2,p} p-antiprism {p}h{}
Wythoff's symbol converts into our own through the filters
pQ2Pq x|o Used if 2 is present
pQrPqRz x|o Used if 2 is not present
To use this, the Wythoff symbol is some permutation of 2 p q | or p q r |.
We rewrite p, q, r in the form P Q R, and then put an x if these fall
before the bar and o if after.
zb pQ2Pq x|o
x3x5o 2 5 | 3
x 5 5 | x goes to p, P goes to 5, 5 is before the bar
x 2 | x goes to 2, 2 is before the bar
3 o | 3 x goes to q, Q goes to 3, 3 is after the bar.
zb pQ2Pq x|o
x3x5o 2 5 | 3
3 5 These are the Wythoff values not equal to 2
x 2 | 2 is before the bar (x): node 2 is x
x 5 5 | 5 is before the bar (x): 5 goes to P, p is x
3 o | 3 3 is after the bar (o): 3 goes to Q, q is o.
Snubs are overloaded: the convention is
sPsQs | p q 2
sPsQsRz | p q r
We have no form for the solitary overload "| p q r s".
It is mayly for a uniform figure to be made of regular polygons, but not
of regular polyhdera. An example is the {3,4,3} antiprism, which has the
following faces.
48 verticies.
2 * x3o4o3o
48 * x3o4o pyramids
192 * x3o3o3o
Of lattices.
1 x4o4o 4 | 4 2 {4,4}
2 x6o3o 3 | 6 2 {6,3}
3 o6o3x 6 | 3 2 {3,6}
4 o6o3o 2 | 6 3 r{3,6}
5 x4x4o 2 4 | 4 t{4,4}
6 x6x3o 2 3 | 6 t{6,3}
7 x6o3x 6 3 | 2 rr{6,3}
8 x6x3x 6 3 2 | tr{5,3}
9 s4s4s | 2 4 4 sr{4,4}
10 s6s3s | 2 3 6 sr{6,3}
11 "xUoLx" triang + squares [has vf 4,4,3,3,3 ]
1 x3x3o3o3z
2 x3o4oAo
3 x3x4oAo
4 x3o4xAo
5 x3x4xAo
6 x4o3o4o
7 o4x3o4o
8 x4x3o4o
9 x4o3x4o
10 o4x3x4o
11 x4x3x4o
12 x4x3o4x
13 x4x3x4x
14 "x3o3o3zLx" octtet + prisms Cubic octet
15 "x3o3o3zLs" octtet + octtet Hexagonal close-pack
16 "x3o3o3zLxs" octtet + prisms, hexagonal close-pack
17 "x4o4oLx" octtet + cubes,
#17 here consosts of a cuboctahedron cut through the square faces. It has
altenating layers of oct-tet trusses and cubes. The vertix figure is a
cuboctahedron cut in half parallel to its square face, and the other half
is a half-octahedron of edge sqrt-2.
It bespeaks of a regular laminahedral class consisting of slices of the
cubic antiprism and cubic layers.
The L branch is added to the simplex ring-group, and supports the s and x
nodes, as follows.
x3o..3zLx Makes prisms of the laminahedral base in alternating rows.
x3o..3zLs reflects the contents of a laminahedra in the face, and so
the band is restored to its original condition.
x3o..3zLxs Take an Ls and insert prisms.
x4o4oLx Layers of cubes and octtet-trusses. Includes square-prism
faces.
1 x3o3o3o3o3z
2 x3x3o3o3o3z
3 x3o3x3o3o3z
4 x3x3x3o3o3z
5 x3x3o3x3o3z
6 x3x3x3x3o3z
7 x3x3x3x3x3z
8 xEo3o3o4x
9 xEo3x3o4x
10 xEo3o3x4x
11 xEo3x3x4x
12 x4o3o3o4o
13 o4x3o3o4o
14 x4x3o3o4o
.context @V4022
:nUniform Polyhedra
.freeze 1
\i\i \b\aUniform Polyhedra\b\v@V4001\v
The numbering of the polytopes is that the same number is used for a figure
and its prism in n+1 dimensions. Thus '7' is used for the dodecahedron and
its prisms in 4D, that prism in 5D and so forth. In 5D one may construct a
dodecahedron-pentagon prism. This is not a line-prism of a 4D figure, and
is therefore allocated a new nr (2D*3D are dealt with under 67).
Wythoff-construction is dropping edges to the mirror-planes of the
fundemental groups. This rule constructs all but a handful of figures.
The number of edges is the number of degrees of freedom.
The snub-figures can be constructed by taking verticies in alternate
regions and constructing a uniform figure. In 3D, this can be done in
every case, there is the special s node that handles this. In four and
higher dimensions this is done only infrequently. The most common case
is snub-r, or halfcube.
An antiprism places a figure and its recriprocal in parallel planes, and
makes faces in the triangular product of matching elements. Apart from
the normal antiprisms, the simplexes yield the cross-polytope x3o..4o, and
there is the antiprism of the x3o4o3o, which has 172 5-cells and 48
octahedral pyramids.
The order of numbering is as follows.
1 Line prisms of the previous dimension
2 A number for each infintie class in the current dimension
3 A number for each dimension combination where the elements are 2+D
4 Figures by Wythoff
a: platonic figures
b: gosset figures having 1 degree and 1 degree vf
c: other wythoff figures, by freedom/symmetry group
5 everything else not yet counted.
P G 1 2 3 4 5 Others
1D 0 1
2D 1 polygons
------------------------------------------------------------------
3D 2 3 7 9 16 18
(3) antiprisms SS 1 1
SQ 1 1 3 1 s3s4s
SF 2 1 3 1 s3s5s
------------------------------------------------------------------
4D 20 21 26 28 31 50 61 65
(21) 2D*2D SSS 1 1 4 2 1
SSQ 1 1 5 3 1 s3s3sAs
SQS 1 1 4 2 1 j5j2j5j
SSF 2 1 1 6 4 1
-------------------------------------------------------------------
5D 67 68 70 71 76 95 114 123 125
(68) 2D*3D SSSS 1 2 6 6 3 1
SSSA 1 3 3 1
SSSQ 1 3 10 10 5 1
-------------------------------------------------------------------
1 2 3 4 5 6 7 8
---------------------------------------------
Prisms 1 1 3 3 4
Wythoff 1 1 15 44 57 152 356 708
Others 3 2
---------------------------------------------
1 1 18 47 57 155 356 712
2 20 68 126 279 635 1347
\bN=1\b
1 Line segment, square, cube, teseract.
\bN=2\b
2 xPo, xPoXx xPoXxXx etc for n=3 or n>4, For n=4 see '1'.
\bN=3\b
3 Antiprism sPs2s; s2s2s = x3o4o; s3s2s = x3o4o
The figuresd by Wythoff-Constructions and the regular SNUB operator.
P x3o3o 4 x3o4o 5 x3o5o 6
x4o3o =1 x5o3o 7
1 o3x4o 8 o5x3o 9
2 x3x3o 10 x3x4o 11 x3x5o 14
x4x3o 12 x5x3o 15
x4o3x 13 x5o3x 16
3 x4x3x 17 x5x3x 18
X s4s3s 19 s5s3s 20
\bN=4\b
21 xPo2xQo The rectangular product of two polygons in 4D.
The figures by Wythoff-construction in 4D
P x3o3o3o 22 x3o3o4o 23 x3o4o3o 24 x3o3o5o 25
x4o3o3o = 1 x5o3o3o 26
G o3x3o3o 27 o3x3o4o = 24 o3x3o5o 28
1 o3o3x4o 29 o3x4o3o 30 o5x3o3o 31
2 x3x3o3o 32 x3x3o4o 36 x3x4o3o 41 x3x3o5o 45
x4x3o4o 37 x5x3o3o 46
o3x3x3o 33 o4x3x3o 38 o3x4x3o 42 o3x3x5o 47
x3o3x3o 34 x3o3x4o = 29 x3o4x3o 43 x3o3x5o 48
x4o3x3o 39 x5o3x3o 49
x3o3o3x 35 x4o3o3x 40 x3o4o3x 44 x3o3o5x 50
3 x3x3x3o 51 x3x3x4o = 41 x3x4x3o 56 x3x3x5o 58
x4x3x3o 53 x5x3x3o 59
x3x3o3x 52 x3x3o4x 54 x3x4o3x 57 x3x3o5x 60
x4x3o3x 55 x5x3o3x 61
4 x3x3x3x 62 x3x3x4x 63 x3x4x3x 64 x3x3x5x 65
The non-mirror-edge figures. These may be found by removing verticies
from x3o3o5o.
s3s4o3o 66 Faces: 24 icosa and 96 tetra Inscribed 24-cell.
j5j2j5j 67 Faces: 20 s2s5s and 260 s2s2s Opposite equators.
\bN=5\b
68 Prisms of any polygon by any figure from 3 to 20.
Figures by the Wythoff-construction in 5D
Figures by Wythoff-Constructions. Numbers in left margin refer to the
number of degrees of freedom.
P x3o3o3o3o 69 x3o3o3o4o 70 x4o3o3o3o =1
G x3o3o3oBo 71
1 o3x3o3o3o 72 o3x3o3o4o 74 o4x3o3o3o 76
o3o3x3o3o 73 o3o3x3o4o 75
2 x3x3o3o3o 77 x3x3o3oBo 83 x3x3o3o4o 86 x4x3o3o3o 92
o3x3x3o3o 78 o3x3x3o4o 87 o4x3x3o3o 93
x3o3x3o3o 79 x3o3x3oBo 84 x3o3x3o4o 88 x4o3x3o3o 94
o3x3o3x3o 80 o3x3o3x4o 89
x3o3o3x3o 81 x3o3o3xBo 85 x3o3o3x4o 90 x4o3o3x3o 95
x3o3o3o3x 82 x3o3o3o4x 91
3 x3x3x3o3o 96 x3x3x3oBo 102 x3x3x3o4o 105 x4x3x3o3o 111
o3x3x3x3o 97 o3x3x3x4o 106
x3x3o3x3o 98 x3x3o3xBo 103 x3x3o3x4o 107 x4x3o3x3o 112
o3x3x3o3x 99 x3o3x3xBo 104 o3x3x3o4x 108 o4x3x3o3x 113
x3x3o3o3x 100 x3x3o3o4x 109 x4x3o3o3x 114
x3o3x3o3x 101 x3o3x3o4x 110
4 x3x3x3x3o 115 x3x3x3xBo 118 x3x3x3x4o 119 x4x3x3x3o 122
x3x3x3o3o 116 x3x3x3o4o 120 x4x3x3x3o 123
x3x3o3x3x 117 x3x3o3x4x 121
5 x3x3x3x3x 124 x3x3x3x4x 125
Other figures.
?126 x3o4o3oJm 24-cell antiprism. This has polygonal faces and
uniform verticies, but the 3-D spaces are not self-uniform.
\bN=6\b
2D*2D*2D 2D = 3
2D*4D 3D = 4-20 inc
3D*3D 4D = 22-66 inc
Figures by Wythoff Construction, by degrees of freedom
Total 1 2 3 4 5 6
-------------------------------------
s 35 3 9 10 9 3 1
h 16 1 4 6 4 1 0
hr 62 5 15 20 15 6 1
g 39 4 9 12 9 4 1
----------------------------
152 13 37 48 37 14 3
j5j2j5j2j5j
-----------
The j5j2j5j is a figure in 4D, consisting of 20 pentagonal antiprisms and
260 simplexes. This has two equators each with 10 antiprisms. Is it
mayly that there is a 6D equal?
This figure is unlikely, since the s5s2s lie in a definite circle in the 4
space, which do not replicate into 6D, it is not as if the contentant
elements lie in 2D but 4D. In any case, the s5s5s lie on a torus of
alternate verticies of a 10*10 ring, and the tetrahedra form in the
antiprism-product. Since the ring in 5,6 axis can not be odd and even,
the figure j5j2j5j2j5j does not exist.
\bN=7\b
2D*2D*3D
2D*5D
3D*4D
Figures by Wythoff's Construction
Total 1 2 3 4 5 6 7
------------------------------------------
s 71 4 12 19 19 12 4 1
h 32 1 5 10 10 5 1
hr 126 6 21 35 35 21 7 1
g 127 7 21 35 35 21 7 1
------------------------------------------
356 18 59 99 99 59 19 3
\bN=8\b
2D*2D*2D*2D
2D*3D*3D
3D*5D
4D*4D
Figures by Wythoff-Constructions (except the cube)
1 2 3 4 5 6 7 8
-----------------------------------------
s 135 4 16 28 38 28 16 4 1
h 64 1 6 15 20 15 6 1
hr 254 7 28 56 70 56 28 8 1
g 255 8 28 56 70 56 28 8 1
----------------------------------------
708 20 78 155 198 155 78 21 3
.context @V4023
:nNew and Old Notation
.freeze 1
\i\i \b\aNotations\b\v@V4001\v
o old style
c old style class This governs the meaning of modifiers
n new style
* see a note on this
- not supported in this system.
\bBranch Actions\b
-------------------------------+----------------------------------------+
o c n | o c n |
-------------------------------+----------------------------------------+
& X direct product | O - - circles and spheres |
A s A second subject | P p * polygo numerator |
B s B third subject | Q n 4 marked branch, 4 |
C s C fourth subject | R n 2 marked branch, 2 |
D p * polygram divisor | Rn p - polygon throughdivisor |
E s E second object | S n 3 unmarked branch, = 3 |
F n 5 marked branch, 5 | T n 5/2 marked branch, 5/2 |
G s G third object | U n U marked branck, infinity |
H n 6 marked branch, 6 | V n 6/2 marked branck, 6/2 |
IJKLMN free for use | WXYZ free for use |
-------------------------------+----------------------------------------+
* The new style differs considerably in the handling of general polygons.
\bNode Actions\b
-------------------------------+----------------------------------------+
o c n | o c n |
-------------------------------+----------------------------------------+
* o unmarked node | : z First loop alias |
/ f x mirror-edge node | :: zz Second loop alias |
\\ f m mirror-margin node | # r laminate mirror |
h/ s snub node | % s Complex node |
v vertix mode | p Petrie node |
f face mode | |
-------------------------------+----------------------------------------+
* The old style does not show unmarked nodes.
* The snub figures are alternate verticies of an all-marked figure
in the old style, this is eg h/S/F/.
\bModifiers\b
The class defines the meaning and type of modifiers that might be applied.
s structural (eg A B C E G). The branch is normally taken to be an
unmarked branch, but it can be altered by the use of a numeric or
polygonal modifier.
old style Aq Af Ah Ap5d2
new style A4 A5 A6 A5/2
applied to % makes the complex node other than 3. eg %S or %fS. The %
is applied to all nodes connected by odd branches (S, F, V, and Podd),
thus %F and %F% are the same thing.
f form (eg / \\ ). This makes a size, normally 1, to be the shortchord
of the polygon named, either n or p class.
old style /&/f
new style (not used)
n numeric. This denotes a specific value. Two modifiers are supported.
Ni generates the complement, ie 1/N + 1/Ni = 1/2.
Nr generates the asterix, ie Nr = PnD(n/2)
p polygon. There are three allowed fields, which reduce according to
the following principle to PpDd.
P8DSDR2 [This is {8,3,4}
P8DSD2R2 All examples of DRr become DrRr
P8D1SD2R2 All examples of D without a number become D1
P8D1SP8D2R2 All Dd acquire the previous Pp.
P8D1SP4D1 All PpDdRr become Pp/hDd/h, h=gcd(p,d,r)
P8SP4 All D1 are removed.
\bExamples\b
Old style New style Name
/SF x3o5o icosahedron 5 | 2 3
/Q/S o3x4x truncated cube 3 2 | 4
S/Q/S o3x4x3o octagonny 3t{3,4,3}
/SSSSB = /4B x3o3o3o3oBo Gosset's 6D figure [27 verticies]
S\Q o3m4o Rhombic dodecahedron
%SS Complex polyhedron 3{3}3{3}3
/S/SSS: x3x3o3o3z Space tiled with tetra + trunc tet
x7o3oA16p Finite map, = Coxeter's "{7,3}_16"
/&/f Golden rectangle
/FVF x5o5/2o5o Second 120-star {5,5/2,5}
.context @V4024
:nThe Special Function
.freeze 1
\i\i \b\aThe Special Function\b\v@V4001\v
The special function is used to derive the circumdiameters of the assorted
one-degree figures. It is derived from the shortchord squares of the
polygons.
S 1 S 3 SS 4 SSS 5 SSSS 6 SSSSS 7 SSSSSS 8 SSSSSSS 9
SA 4 SSA 4 SSSA 4 SSSSA 4 SSSSSA 4 SSSSSSA 4
S& 6 SSB B SSSB 4 SSSSB 3 SSSSSB 2 SSSSSSB 1
Q 2 Q 2 SQ 2 SSQ 2 SSSQ 2 SSSSQ 2 SSSSSQ 2 SSSSSSQ 2
SQS 1
H 3 H 1 SH 0
The circumdiam of a figure is twice the special function of the vertix
figure, divided by the vfd of the total.
Thus, for /QSSSSS, the cd is 2[SSSSS]/[QSSSSS] = 2*7/2 = 7
for QSSSS/S the cd is 2[QSSS][.]/[QSSSSS] = 2 2 2 / 2 = 4
for SS/SSB the cd is 2[S][S][.]/[SSSSB ] = 2 3 3 2 / 3 = 12
for SSS/SS the cd is 2[SS][S] /[SSSSS] = 2 4 3 / 7 = 2:5151
The special function can be derived iteratively, as follows.
1
2
p 4-p
q 8-2p-2q 2(4-p) - q(2)
r 16-4p-4q-4r+pr 2(8-2p-2q) - r(4-p)
s 32-8p-8q-8r-8s+2ps+2qs+2pr
For special cases
p p3 p33 p333 p3333 p3..34 633 5333
p3.. 3-b 4-2b 5-3b 6-4b 7-5b 2-2b 534 634 5334
3p 3p3 3p33 3p333 3p3..34 353 363 34333
4-2b 5-4b 6-6p 7-8b 2-4b 344 3434 34334
33p 33p3 33p33 33433
5-3b 6-6b 7-9b
pp p3p p33p p333p .... 535 636 5335
444 635
For branches radiating from a common centre, the figrue is
2^(n-2)(4-a-b-c...)
SSA 4(4-1-1-1) = 4
QSA 4(4-1-1-2) = 0
FSA 4(4-f-1-1) = -0.618033
HSA 4(4-3-1-1) = -4
SSAB 8(4-1-1-1-1) = 0
QSAB 8(4-2-1-1-1) = -8
ESSSA 16(4-1-1-1-1-1) =-16
.context @V4031
:nRegular Finite Maps
.freeze 1
\i\i \b\aFinite Regular Maps\b\v@V4001\v
{3,3}_4 x3o3oA4p | 4 6 4 | 24 | 2 0
{4,3}_3 x4o3oA3p | 4 6 3 | 24 | 1 --
{4,3}_6 x4o3oA6p | 8 12 6 | 48 | 2 0
{6,3}_4 x6o3oA4p | 8 12 4 | 48 | 0 1
{6,4}_3 x6o4oA3p | 6 12 4 | 48 | -2 --
{5,3}_5 x5o3oA5p | 10 15 6 | 60 | 1 --
{5,5}_3 x5o5oA3p | 6 15 6 | 60 | -3 --
{4,4}_4 x4o4oA4p | 8 16 8 | 64 | 0 1
{6,3}_6 x6o3oA6p | 18 27 9 | 108 | 0 1
{6,6}_3 x6o6oA3p | 9 27 9 | 108 | -9 --
{5,3}_10 x5o3oA10p | 20 30 12 | 120 | 2 0
{10,3}_5 x10o3oA5p | 20 30 6 | 120 | -4 --
{10,5}_3 x10o5oA3p | 12 30 6 | 120 | -12 --
{5,4}_5 x5o4oA5p | 20 40 16 | 160 | -4 --
{5,5}_4 x5o5oA4p | 16 40 16 | 160 | -8 5
{6,3}_8 x6o3oA8p | 32 48 16 | 192 | 0 1
{8,3}_6 x8o3oA6p | 32 48 12 | 192 | -4 3
{8,6}_3 x8o6oA3p | 16 48 12 | 192 | -20 --
{5,4}_6 x5o4oA6p | 30 60 24 | 240 | -6 4
{6,4}_5 x6o4oA5p | 30 60 20 | 240 | -10 --
{6,5}_4 x6o5oA4p | 24 60 20 | 240 | -16 9
{4,4}_8 x4o4oA8p | 32 64 32 | 256 | 0 1
{8,4}_4 x8o4oA4p | 32 64 16 | 256 | -16 9
{6,3}_10 x6o3oA10p | 50 75 25 | 300 | 0 1
{10,3}_6 x10o3oA6p | 50 75 15 | 300 | -10 6
{10,6}_3 x10o6oA3p | 25 75 15 | 300 | -35 --
{7,3}_8 x7o3oA8p | 56 84 24 | 336 | -4 3
{8,3}_7 x8o3oA7p | 56 84 21 | 336 | -7 --
{8,7}_3 x8o7oA3p | 24 84 21 | 336 | -39 --
{4,4}_10 x4o4oA10p | 50 100 50 | 400 | 0 1
{10,4}_4 x10o4oA4p | 50 100 20 | 400 | -30 16
{6,3}_12 x6o3oA12p | 72 108 36 | 432 | 0 1
{12,3}_6 x12o3oA6p | 72 108 18 | 432 | -18 10
{12,6}_3 x12o6oA3p | 36 108 18 | 432 | -54 --
{7,3}_9 x7o3oA9p | 84 126 36 | 504 | -6 --
{9,3}_7 x9o3oA7p | 84 126 28 | 504 | -14 --
{9,7}_3 x9o7oA3p | 36 126 28 | 504 | -62 --
{4,4}_12 x4o4oA12p | 72 144 72 | 576 | 0 1
{12,4}_4 x12o4oA4p | 72 144 24 | 576 | -48 25
{6,3}_14 x6o3oA14p | 98 147 49 | 588 | 0 1
{14,3}_6 x14o3oA6p | 98 147 21 | 588 | -28 15
{14,6}_3 x14o6oA3p | 49 147 21 | 588 | -77 --
{5,5}_5 x5o5oA5p | 66 165 66 | 660 | -33 --
{8,3}_8 x8o3oA8p | 112 168 42 | 672 | -14 8
{8,8}_3 x8o8oA3p | 42 168 42 | 672 | -84 --
{6,3}_16 x6o3oA16p | 128 192 64 | 768 | 0 1
{16,3}_6 x16o3oA6p | 128 192 24 | 768 | -40 21
{16,6}_3 x16o6oA3p | 64 192 24 | 768 | -104 --
{4,4}_14 x4o4oA14p | 98 196 98 | 784 | 0 1
{14,4}_4 x14o4oA4p | 98 196 28 | 784 | -70 36
{6,3}_18 x6o3oA18p | 162 243 81 | 972 | 0 1
{18,3}_6 x18o3oA6p | 162 243 27 | 972 | -54 28
{18,6}_3 x18o6oA3p | 81 243 27 | 972 | -135 --
{4,4}_16 x4o4oA16p | 128 256 128 | 1024 | 0 1
{16,4}_4 x16o4oA4p | 128 256 32 | 1024 | -96 49
{7,3}_13 x7o3oA13p | 182 273 78 | 1092 | -13 --
{13,3}_7 x13o3oA7p | 182 273 42 | 1092 | -49 --
{13,7}_3 x13o7oA3p | 78 273 42 | 1092 | -153 --
{4,4}_18 x4o4oA18p | 162 324 162 | 1296 | 0 1
{18,4}_4 x18o4oA4p | 162 324 36 | 1296 | -126 64
{5,4}_8 x5o4oA8p | 180 360 144 | 1440 | -36 19
{8,4}_5 x8o4oA5p | 180 360 90 | 1440 | -90 --
{8,5}_4 x8o5oA4p | 144 360 90 | 1440 | -126 64
{7,3}_12 x7o3oA12p | 364 546 156 | 2184 | -26 14
{12,3}_7 x12o3oA7p | 364 546 91 | 2184 | -91 --
{12,7}_3 x12o7oA3p | 156 546 91 | 2184 | -299 --
{7,3}_14 x7o3oA14p | 364 546 156 | 2184 | -26 14
{14,3}_7 x14o3oA7p | 364 546 78 | 2184 | -104 --
{14,7}_3 x14o7oA3p | 156 546 78 | 2184 | -312 --
{9,3}_9 x9o3oA9p | 570 855 190 | 3420 | -95 --
{9,9}_3 x9o9oA3p | 190 855 190 | 3420 | -475 --
{8,3}_10 x8o3oA10p | 720 1080 270 | 4320 | -90 46
{10,3}_8 x10o3oA8p | 720 1080 216 | 4320 | -144 73
{10,8}_3 x10o8oA3p | 270 1080 216 | 4320 | -594 --
{6,4}_7 x6o4oA7p | 546 1092 364 | 4368 | -182 --
{7,4}_6 x7o4oA6p | 546 1092 312 | 4368 | -234 118
{7,6}_4 x7o6oA4p | 364 1092 312 | 4368 | -416 209
{5,4}_9 x5o4oA9p | 855 1710 684 | 6840 | -171 --
{9,4}_5 x9o4oA5p | 855 1710 380 | 6840 | -475 --
{9,5}_4 x9o5oA4p | 684 1710 380 | 6840 | -646 324
{8,3}_11 x8o3oA11p | 2024 3036 759 | 12144 | -253 --
{11,3}_8 x11o3oA8p | 2024 3036 552 | 12144 | -460 231
{11,8}_3 x11o8oA3p | 759 3036 552 | 12144 | -1725 --
{7,3}_15 x7o3oA15p | 2030 3045 870 | 12180 | -145 --
{15,3}_7 x15o3oA7p | 2030 3045 406 | 12180 | -609 --
{15,7}_3 x15o7oA3p | 870 3045 406 | 12180 | -1769 --
{7,3}_16 x7o3oA16p | 3584 5376 1536 | 21504 | -256 129
{16,3}_7 x16o3oA7p | 3584 5376 672 | 21504 | -1120 --
{16,7}_3 x16o7oA3p | 1536 5376 672 | 21504 | -3168 --
.context @V4025
:nA New Order
.freeze 1
\i\i \b\aA New Order\b\v@V4001\v
This is the proposed order for the convex uniform figures and honeycombs.
Rule 1. The rectangular product preserves uniformity. Since there is
a one-dimensional uniform figure, then the same number is used for the
line-prism of a figure as the figure itself. For example, a pentagon
is counted as a member of the second uniform figure (polygons). The
pentagonal prism is counted therefore as one of the second uniform
polyhedra.
Rule 2. There are prism-products of figures with more than one dimension.
For example, in 4 dimensions, the bihexagon prism is the product of two
hexagons. These are counted as a separate entry for each combination of
numbers, in ascending order, that are greater than two, and sum to the
dimension. An entry 3d*2d means the rectangular product of a simple 2d
figure and a simple 3d figure, as described in rule 3.
Rule 3. All figures that are not prisms of lesser figures are designated
as simple figures. The vast majority of these fall through applying
Wythoff's mirror-edge construction to the native symmetries of that
dimension.
a. Present the symmetries in a reduced form
b. Apply Wythoff's construction to the reduced form
c. Apply any other regular construction to the symmetries
Currently, sPsQs.
d. Append any figures not found by b. and c.
e. Identify and remove from the count, all of the same shape.
f. Remove all that are prisms (ie x4o[3o] and x4o3[o3]o4v.
In this table, the line `Previous' is the number of line-prisms or aperigon
prisms. Prisms are products of 2d or greater figures. New are those other
than listed before, and Total is the sum. For the parabolic case, the
prisms are calculated in full: 55 = triangle(10). This is because the
parabolic case has no infinite sets.
1e 2e 3e 4e 1p 2p 3p 4p
Previous - 1 2 20 - 1 11 26
Prisms - - - 1 - - - 55
New 1 1 18 46 1 10 15 57
Total 1 2 20 67 1 11 26 138
In rule a, the values of some of the nodes are pre-set to x or o, and do
not vary. This table identifies all other variations.
xPvPo aPvPa = vPa4o for a=x, o
-3xAo = -3oAx, -3aAa = -3a4o.
oEx3- = xEo3-, aEa3- = o4a3-.
For rule b, v can freely be x or o. But note, that if the figure
`reverses' to a previously counted member, it is not counted again.
Thus x3x3o3o is the same as o3o3x3x. These are indicated as E
(End-symmetric).
In rule c, the sole example to date is sPsQs. One can apply the rule s
(which removes every second vertix from an x form) to any dimension. The
vertex figure is a simplex, with triangular(n-1) sides and n degrees of
freedom. This equates for all n=3, and for a scattering of higher
figures, which are dealt with as irregulars.
Rule d consists of a handful of fugures not directly derivable from
Wythoff construction.
s3s3sAs Remove vertices of inscribed x3o4o3o from x3o3o5o.
j5j2j5j Remove two opposite decagons of x3o3o5o.
sEs3s3sAs Divide edges of x3o4o3o3o in ratio 1:phi.
Laminae Divide x3o3o3o[3o]z into layers with parallel planes.
P insert a prism of unit height based on the planes.
R Use plane as mirrors ( not different in 2D)
PR Use planes as mirrors and insert prisms.
---------------------------------------------------------------------------
The elliptic cases to four dimensions.
Form id res Identities
1d x 1 - 1 1
2d xPo 1 * 1 2 2d [1] x4o
3d x3v3o S 3 1 2 3 3d [1] x4o3o
v3v4v S 8 1 7 3 x3o5o s3s3s
v3v5v S 8 - 8 3 4d [1] x4o3o3o
4d 2D * 2D 1 - 1 P x3o4o3o o3x3o4o
v3v3v3v E 9 - 9 4 o3x4o3o x3o3x4o
v3v3v4v 15 4 11 4 x3x4o3o x3x3x4o
v3v4v3v E 9 - 9 4 s3s4o3o s3s3s4o
v3v3v5v 15 - 15 4
s3s3sAs 1 - 1 4
j5j2j5j 1 - 1 4
The parabolic figures to 4 dimensions as follows.
1d xUo 1 - 1 1 Identities
2d x4v4o S 3 1 2 2 2d [1] x4o4o
v3v6v S 8 1 7 2 x6o3o x3x6o
Laminae 1 - 1 2 3d [1] x4o3o4o x4o3o4x
3d v4v3v4v E 9 2 7 3 4d [1] x4o3o3o4o x4o3o3o4x
v4v3xAo 4 - 4 3 x3o4o3o3o o3o4o3x3o o4o3x3o4o
x3x3o3o3z 1 - 1 3 o3x4o3o3o o3o4x3o3x o4x3o3x4o
Laminae 3 - 3 3 x3x4o3o3o o3o4x3x3x o4x3x3x4o
4d 2D * 2D 55 - 55 P s3s4o3o3o o3o4s3s3s o4s3s3s4o
v3v4v3v3v 31 3 28 4
v4v3v3v4v 19 5 14 4
x4v3v3xAo 4 - 4 4
v3v3v3v3v3z 7 - 7 4
Laminae 3 - 3 4
s3s4o3o3o 1 - 1 4
.context @V4026
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3-dimensions
------------
x-o-o o-x-o N r
x-x-o x-x-x truncated N t tr
x-o-x rhombi-N rr
s-s-s Snub s
-3-3- tetrahedron T
-3-4- octahedron O
-4-3- cube C
-4-3- cubocahedron CO
-3-5- icosahedron I
-5-3- dodecahedron D
-5-3- icosadodecahedron ID
4-dimensions
------------
x-o-o-o N
o-x-o-o rectified N r
x-x-o-o truncated N t
x-o-x-o cantellated N c
x-o-o-x runcinated N rr
o-x-x-o bitruncated N tt
x-x-x-o cantitruncated N ct
x-x-o-x runcitruncated N rt
x-x-x-x omnitruncated N ot
-3-3-3- 5-cell pentachoron
-3-3-4- 16-cell hexadecachoron
-4-3-3- 8-cell octachoron tesseract
-3-4-3- 24-cell icosatetrachoron
-3-3-5- 600-cell hexacosichoron
-5-3-3- 120-cell hecatonicosachoron
s3s4o3o snub 24-cell
j5j2j5j grand antiprism
eg
x-o-o-o N x-x-o-o truncated
o-x-o-o rectified
o-x-o-o bitruncated
x-o-x-o cantellated x-x-x-o cantitruncated
x-o-o-x runcinated x-x-o-x runcitruncated
x-x-x-x omnitruncated
Revised names
x=3d x=4d
x-o-o x-o-o-o N
x-x-o x-x-o-o truncated
o-x-o-o bitruncated
o-x-o o-x-x-o mesotruncated
x-o-x-o cantellated
x-x-x-o cantitruncated
x-o-x x-o-o-x rhombitruncatd
x-x-o-x
x-x-x x-x-x-x omnitruncated
s-s-s snub
.context @V4027
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\bRevised Numbers and Names\b
The numbers will continue to preserve across line prisms, as follows.
1 line segment
2 polygons, except p=4
{4} = (1)
\b3D figures\b
1 cube, as rectangular prism
2 all other polygonal prisms square prism = (1)
3 polygonal antiprisms digon ~ = (4), triangle ~ = (5)
Platonic or regular figures
4 tetrahedron
5 octahedron
(1) cube
6 icosahedron
7 dodecahedron
Archimedian figures
8-20 See table of names
\b4D figures\b
1-20 line prisms of 3D figures
21 rectangular product of two polygons
Platonic figures
22 x3o3o3o pentachoron
23 x3o3o4o hexadecachoron
(1) x4o3o3o tesseract, octachoron
24 x3o4o3o tetraicosachoron
25 x3o3o5o centachoron
26 x5o3o3o pentacentachoron.
27-65 Whytoff-constructs
66 s3s4o3o Snub 24-cell
67 j5j2j5j great antiprism
333 334 433 343 335 533 33 34 43 35 54
xooo 5 16 8 24 600 120 -cell T O C I D
----------------------------- -----------------
xooo 22 23 (1) 24 25 26 4 5 (1) 6 7 xoo
oxoo 27 (24) 28 29 30 31 rectified (5) -> 8 9 <- oxo
xxoo 32 33 34 35 36 37 truncated 10 11 12 13 14 xxo
oxxo 38 39 (39) 40 41 (41) bitruncated
xoxo 42 (29) 43 44 45 46 cantellated
xoox 47 48 (48) 49 50 (50) runcinated (8 ) 15 <- 16 <- xox
xxxo 51 (35) 52 53 54 55 cantitruncated
xxox 56 57 58 59 60 61 runcitruncated
xxxx 62 63 (63) 64 65 (65) omnitruncated (11) 17 <- 18 <- xxx
ssoo 66 snub (6) 19 20 sss
.context @V4028
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\bSymbols\b
This is designed to give the figures shorter names than the fully expanded
Dynkins-symbol. This symbol is comprised of three elements, the dimension,
the parent regular figure, and the mode of change.
In this regard, it can be regarded as something akin to, say, tT or rrCO,
except that the naming is more akin to the Dynkin symbol.
The general form of a symbol is, eg Di3. The letter D designates a 4D figure
since D is the fourth letter. Di designates a sequence _3_3_5_, the 4D
analogy of the icosahedron, and 3 is in binary 000011, which is reversed and
written in o and x, as xxoooooo, ie Di3 is x3x3o5o.
Prisms are by conjuction. A dodecahedron is Cd1, a line segment is A, and
thus a dodecagonal prism is ACd1.
A 1 dimensional figures
B 2 dimensional figures - polygons except the square
C 3 dimensional figures
Ca antiprisms
BB 4 dimensional figures - prisms of two polygons
D 4 dimensional figures - other elements.
B C D
t -3- -3-3- -3-3-3- tetrahedron 5-cell
o -4- -3-4- -3-3-4- octahedron 16-cell
c -4- -4-3- -4-3-3- cube tesseract
q -3-4-3- 24-cell
i -5- -3-5- -3-3-5- icosahedron 600-cell
d -5- -5-3- -5-3-3- duodecahedrom 120-cell
1 x-o x-o-o x-o-o-o - -
2 o-x-o o-x-o-o rectified rectified
3 x-x x-x-o x-x-o-o truncated truncated
6 o-x-x-o bitruncated
5 x-o-x x-o-x-o runciated cantellated
9 x-o-o-x runciated
7 x-x-x x-x-x-o omnitruncated cantitruncated
11 x-x-o-x runcitruncated
15 x-x-x-x omnitruncated
8 s-s-s snub
16 s-s-o-o snub
While in 3 and 4 dimensions, it is useful to use names like this, in five
dimensions, there are 19 grades and three figures only. In 6, 7 and 8
dimensions, there are figures where these grades do not apply: the
\bUniform Figures enumerated\b
In one dimension there is only the line segment.
1 A The nth power of a line segment, being squares, cubes,
tesseracts and so forth.
In two dimensions, there is a line-prism, and other polygons
1 A Square - as square of line segment.
2 B All other polygons
In three dimensions, there are line prisms of 1-2, and the following.
1 A Cube as cube of line segment
2 B polygonal prisms
3 C Antiprisms.
4-20 C Other figures; note the cube at (1)
In four dimensions, there are line-prisms of 1-20.
1 A Tesseract, as cube-prism or biquadrate of line segment.
2 B polyogon-square prisms
3-20 C line - prism of 3D figure
21 BB polygon-polygon prisms
22-67 D Other 4D figures - note tesseract at (1).
In five dimensions, there are line-prism of 1-67, and more figures.
1 A 5D cube
2 B polygon-cube prisms
3-20 C polyhedron-square prisms
21 BB polygon-polygon-line prisms
22-67 D polychora prisms
68 BC polygon-polyhedra prisms
69-125 E new 5-dimensional figures.
\bConsolidated List\b
The order of this list is the same in all dimensions.
------[ 1 ]-----------------------------------------------------
1 line, square, cube, tesseract, or any higher power of a line
When refered to in a product, includes the figure itself.
Thus, under (2), we refer to pologon * any of (1), the group (2)
includes the polygons themselves.
------[ 2 ]-----------------------------------------------------
2 polygon (except square) * any of (1)
There are an infinite number of these. We do not count 1, 2, or 4
in this list, since
1 does not exist
2 is a back-to-back line, and thus does not have volume
4 is the square, which is an element of 1.
------[ 3 ]-----------------------------------------------------
3 antiprism (for p > 3) * any of (1).
the line-antiprism is a tetrahedron (#4)
the triangle-antiprism is a octahedron (#5)
The group of antiprisms has meaning in all dimensions, being a line
prism whose opposite faces are duals, and sides are triangular products
of matching elements. Because the opposite faces are duals, they are
usually different. This list is all the antiprisms of the self-duals.
polygons = figure as listed here
simplexes = cross polytopes ie -at1 = -o1.
3 4 3 = this yields faces that are not uniform - octahedral pyramids.
4-20 These 3D figures * any of (1).
* # Ct Co Cc Ci Cd
33 34 43 35 53 * *
T O C I D *
------------------------
1 4 4 5 =1 6 7 xoo
2 =5 8 < 9 < r oxo
3 6 10 11 12 13 14 t xxo
5 =8 15 < 16 < rr xox
7 =11 17 < 18 < tr xxx
8 =6 > 19 > 20 s sss
The snub derives by removing half the verticies of the omnitruncate, and
replacing them with simplexes. Since the omnitruncate has n degrees of
freedom, and the simplex has (n)(n-1)/2 edges, when the simplex has more
edges than verticies, the snub figure is generally not uniform. In
three dimensions, there are three verticies and three edges, so a solution
is always available. In four dimensions, there are four verticies and
six edges, so a solution rarely happens.
The only examples past three dimensions are of the form s3s4o[3o].
------[ 4 ]-----------------------------------------------------
21 any of (2), by any equal or lesser of (2) by any of (1)
For this purpose, the square is not a polygon. The order runs 3*3, 5*3
5*5, 6*3, 6*5, 6*6 &c.
22-67 These 4D figures, by any of (1)
* # Dt Do Dc Dq Di Dd
* 333 334 433 343 335 533 *
* 5 16 8 24 600 120 -cell
--------------------------------
1 8 22 23 =1 24 25 26 xooo
2 4 r 27 =24 28 29 30 31 oxoo
3 12 t 32 33 34 35 36 37 xxoo
6 tt 38 > 39 40 > 41 oxxo
5 10 c 42 =29 43 44 45 46 xoxo
9 rr 47 > 48 49 > 50 xoox
7 14 ct 51 =35 52 53 54 55 xxxo
11 13 rt 56 57 58 59 60 61 xxox
15 ot 62 > 63 64 > 65 xxxx
16 - - - 66 67 -
The figures 66 and 67 are not derived by Wythoff-construction. They can
be had from the 600-cell as follows.
66 s3s4o3o snub 24-cell icosahedra inscribed 24-cell
67 j5j2j5j grand antiprism penta antiprism opposite decagons
The s3s4o3o (67) is derived by dividing the edges of the 24-cell into the
ratio 1 to 1.618033... The octahedral faces become icosahedra, and there
are tetrahedral faces at each old vertex. The edges become in part, the
elevations of 96 further tetrahedra. Gosset errected icosahedral pyramids,
caps of 20 tetrahedra, to yield the 600-cell. This has 20*24+96+24 faces
and 96+24 verticies.
The grand antiprism is derived by removing the two decagons that lie in
the wx, yz planes. The original vertix becomes a pentagonal antiprism,
this is the shape resulting when opposite vertices of the icosahedron are
removed. The figure is an example of a laminahedron, which had different
nature of faces on opposite sides of a surface.
------[ 5 ]-----------------------------------------------------
68 Any polygon by any of the figures 3-20 by any figure at (1)
The order runs 3 * any of 3-20, 5 * any of 3-20, and so forth.
69-125 Any of these 5D figures, by any of (1)
3333 E333 3334 4333
* # Et Eh Eo Ec
--------------------------
1 16 69 70 =1 xoooo
2 8 71 72 73 74 oxooo
4 75 76 < ooxoo
3 24 77 78 79 xxooo
6 12 80 81 82 83 oxxoo
5 20 84 85 86 xoxoo
10 87 88 89 < oxoxo
9 18 90 91# 92 93 xooxo
17 94 > 95 xooox
7 28 96 97 98 xxxoo
14 99 100 101 < oxxxo
11 26 102 103# 104 105 xxoxo
13 22 106 107# 108 109 xoxxo
19 25 110 111 112 xxoox
21 113 114 < xoxox
15 30 115 116# 117 118 xxxxo
23 29 119 120 121 xxxox
27 122 123 < xxoxx
31 124 125 < xxxxx
19 7 19 11 = 56
The h group appears in this and all later dimensions. It designates the
halfcubic group, figures derived from removing alternate vertices of the
cube equivalant. While all possible numbers of this group are permitted,
most are duplicates, and only double-odd numbers yield anything different.
Let x0, x1, x2, x3 be 4a+0,1,2,3 respectively.
-hx0 = -cx0 eg Fh12 = Fc12
-hx1 = -hx2 eg Fh13 = Fh14
-hx2 = distinct eg Fh14 = Fh14
-hx3 = -cx2 eg Fh15 = Fc14
In the case of -hx0 and -hx3, these may resolve to simpler forms in -o.
------[ 6 ]-----------------------------------------------------
.context @V4029
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Coxeter style names
xooo 0
oxoo 1 r r rectified runcic 433
xxoo 0,1 t t truncated
oxxo 2,3 2t tt bitruncated runcicantic 433
ooxx 3,4 cantic 433
xoxo 0,2 c cantellated
xoox 0,3 rr runciated
xxxo 0,1,2 ct cantitruncated
xxox 0,1,3 rt runcitruncated
xxxx 0,1,2,3 ot omnitruncated
ssoo s s snub 343
John H Conway
333 5- penta-
334 16- hexa[kai]deca-
433 8- octa-
343 24- icosatetra- polyoctahedron
335 600- hexakosi[oi]- polytetrahedron
533 120- hecatonicosa- polydodecahedron
num- -cell most commonly used
num- greek- -hedroid Henry Parker Manning
num- greek- -tope For general n-dimensional
num- greek -choron
Norman W Johnson, H S M Coxeter, uses `-cell' names.
Role names
x3o3o3o simplex
x4o3o3o tesseract
x&x&x&x measure polytope, orthotope
x4o3o&x cube [hyper]prism
x4o&x4o square duoprism, square hyperprism
x3o3o4o cross polytope
s4o3o3o Half-tesseract, Half-octachoron
x3o3o AP tetrahedral antiprism
x3o4o3o
o3m3o4o "Four-dimensional rhombic-dodecahedron analogue"
x3x3o5o "Four-dimensional soccer ball"
Names by Jonathan Bowers
333 433 343 533
xooo 1 Pen 10 Tes 22 Ico 32 Hi
oxoo 2 Rap 11 Rit 23 Rico 33 Rahi
ooxo =22 34 Rox
ooox 12 Hex 35 Ex
xxoo 3 Tip 13 Tat 24 Tico 36 Thi
xoxo 4 Srip 14 Srit 25 Srico 37 Srahi
xoox 5 Spid 15 Sidprith 26 Spic 38 Sidpixhi
oxxo 6 Deca 16 Tah 27 Cont 39 Xhi
oxox =23 40 Srix
ooxx 17 Thex 41 Tex
xxxo 7 Grip 18 Grit 28 Grico 42 Grahi
xxox 8 Prip 19 Proh 29 Prico 43 Prix
xoxx 20 Prit 44 Prahi
oxxx =24 45 Grix
xxxx 9 Gippid 21 Gidpith 30 Gippic 46 Gidpixhi
sooo =12
soxo =17
soox =11
soxx =16
ssoo 31 Sadi
j5j2j5j 47 Gap
Names by
333
1 xooo pentachoron
2 oxoo dispentachoron
3 xxoo truncated pentachoron
4 xoxo small prismatodispentachoron
5 xoox small prismatodecachoron
6 oxxo decachoron
7 xxxo great prismatodispentachoron
8 xxox diprismatodispentachoron
9 xxxx great prismatodecachoron
433
10 xooo tesseract
11 oxoo tesseractihexadecachoron
12 ooox hexadecachoron
13 xxoo truncated tesseract
14 xoxo small prismatotesseractihexadecachoron
15 xoox small diprismatotesseractihexadecachoron
16 oxxo truncated-octahedral tesseractihexadecachoron
17 ooxx truncated hexadecachoron
18 xxxo great prismatotesseractihexadecachoron