Hi people
Here's my notation, given in brief.
It's fairly long, because the notation
has grown quite
vigorously in the last 30 years. Might give you a clue on
what I'm talking about.
Wendy
-------------------------------------------------------------------
Abstract:
---------
General description of my
notion and notations for polytopes. The notation
was based
on the Dynkins symbol, but has outgrown that to include spheres,
exotic prisms, and other interesting bits.
Wendy Krieger
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
The Notion and Notation
The basis of my notation is
the Dynkins symbol, to which I have applied a
rather extensive
termology. Once the Dynkins symbol has been absorbed, it
is further extended to include many other things.
The
notation is one of a construction, not an outcome. By itself,
it would
not resolve the cube from the square prism. My
first encounter with the
24-cell was as o3m3o4o, and while I
correctly describe its face and
everything, (the face is
m&m4o, the dual of x&x4o, I did in effect fail
to
identify the square dipyramid as an octahedron.
Also, there
is also a fair amount of realisation on what the thing actually
means. I have had a rather lively discussion on what
x10/2o should mean.
Constructions
-------------
The Schaffli construction consists of arranging polygons, q
at a corner.
This produces the regular polyhedra. These
are placed r at an edge, which
produces a polychora. The
symbol consists simply of these numbers.
2D
{p} p-gon
3D
{p,q} p-gon, q at a
corner there are 5
of these
4D {p,q,r} p,q hedron, r at an
edge there are 6 of these
5D
{p,q,r,s} p,q,r-chora, s at a margin. for this and higher, 3
only.
&c.
The Coxeter-Stott construction consists of
expanding, contracting, and
truncating the regular
figures. The result is denoted as xx{p,q,r}, where
xx is
the distortion. All of these operators have distinct names,
such
as "truncated" t- or "rectified" r-.
In the small
dimensions, there are relatively few operators, each with a
fairly good yield of figures. The six operators in 3D
yield 18 figures,
and the 9 operators in 4D yield 44
figures. A catchall operator 'ss' can
be used so that the
remaining pair are picked up. The snub{3,4,3} is the
ss{3,4,3}, and the grand antiprism is the ss{3,3,5}. The
operator s has
an entirely different meaning.
The number
of operators in five and higher climb quickly, the yield drops
drastically. In higher dimensions, a yield of 3 per
operator is quite good.
Whole classes yield one new figure, or
two.
The Wythoff construction consists of creating
mirror-edges. A mirror-edge
is an edge perpendicularly
bisected by a mirror. Generally, the vertex
can be in a
huge wedge-spaced region radiating from the centre, and there
are as many degrees of freedom as there are sides to this
wedge. A point
can be in the interior or on one or more
walls.
This construction yields 62 of the 67 uniform figures
to 4D, and presumably
all of those in higher dimensions.
Of the remaining 5, 4 are produced by
considering alternating
verticies (which also leads to alternate
constructions of other
figures as well), and the remaining #67 grand
antiprism is
handled differently.
Wythoff's notation is the fairly
intuitive device of naming the mirrors by
the opposite angle,
and using an off|on style notation. A mirror called p
is
opposite an angle of pi/p radians.
The remaining cases are
the snub class, which are alternating verticies of
the
omnitruncate. The omnitruncate has three degrees of freedom,
and so it
is possible to set it so an equalateral triangle forms
when alternating
verticies are removed. All other faces
are therefore the same. In this
case, the symbol | p q r
(which properly means a point on all three mirrors,
ie a point
at the centre), is overloaded to mean a snub figure.
In four
dimensions, the snub operator still operates on the omnitruncate,
but there are four degrees of freedom and six edges, so in most
cases, the
six edges can not be set equal. The snub{3,4,3}
is actually alternate
verticies of some form of the
truncated{3,4,3}, not the omnitruncate.
In four dimensions,
the naming convention disappears, as the mirrors are
no longer
opposite dihedra but verticies. If the mirrors are named, the
thing will still work. But one needs to know more than the
face names,
because the {p,q,p} only has two different
face-shapes say A and B. The
notation A B | B A means either the
truncate or the cantellated {p,q,p}.
THE DYNKINS SYMBOL
------------------
This uses the unintuitive use of the
dual of the fundemental region. The
important features in
this is the mirrors and the dihedral angles, the
mirrors held
apart by the dihedral angles. Because of this, it provides
a perfectly general notation for high and low dimensions alike,
and much
use comes from it. But to make this happen, one
must understand how the
thing works.
The diagram at left represents the fundemental
5 region of the {3,3,5}. It has four
mirrors, which
o----o-----o-----o are represented by
circles "o". I have given these
a
b c d
letter-names, a-d. Neither the letters or these
names are part of the symbol, but just something
to allow us to
talk about it. That's all it does.
The NODES a, b, c,
d represent mirrors.
The BRANCHES are the lines connecting
nodes. Branches are drawn only for
angles other than a
right-angle. There is no direct connection between
nodes a
and c, so these mirrors are at right angles.
Two nodes have
a DIRECT CONNECTION if there is a branch joining them, and
an
indirect connection if there is a series of branches connecting
them.
Nodes a and c are connected via b.
The VERTEX NODE
is directly connected to every node that a non-zero
mirror-edge
drops to. This is usually shown by ringing the node, either
as (o) or @. For example, the cantellated {3,3,5} is as
shown below.
There are two kinds of edges, one to "a" and one to "c".
5 What sort of faces form, is decided by
removing each of
@---o---@---o the four nodes in
turn, leaving the other three stand.
a b
c d This can go all the way down. The
shared margins are
found by removing pairs of nodes.
Faces
Margins
a b c
d
a b c d
.
o---@-5-o
icosadodecahedron
. . @-5-o pentagon
@ . @-5-o pentagonal
prism
. o . o *
[point]
@---o . o *
[no
face]
. o---@ . triangle
@---o---@ . cuboctahedron =
rr{3,3} @ . .
o * [line]
@ . @ . square
@---o . . triangle
There
is no face at node 3, since the vertex node is no longer connected
to node "d". Even so, it still shares a margin with "d",
since removing
both nodes leave the balance connected to the
vertex figure. In this
case, the face at node c is a
triangle-prism of zero height, the base and
top is shared with
d: removing this zero-height prism leaves the face at
d to share
the triangle with another face at "d".
NOTIONS to assist in
writing the symbol
---------------------------------------
The regular Dynkins symbol presents no problem. One
needs a symbol for
each possible branch that occurs, and for
each state of node.
Nodes
o @ () -3- -4-
-5- -6-
/
S Q F
H Old style (High dimensions)
o x s
3 4 5
6 New style (Low dimensions)
@-3-o-3-@-5-o = c{3,5}
Old / S S / F =
/SS/F = /2/F
New x 3 o 3 x 5 o = x3o3x5o
s-3-s-5-s = s{5,3}
Old h / S / F / = h/S/F/
New s 3 s 5 s = s3s5s
The older system is designed for higher
dimensions. One can replace
great slices of "S" branches
by a count of them. So "SSSSSSSSSS" can be
written as
"10". If a mark occurs inside this, it is shown, eg
"SSSS/SSSSSS"
as "4/6". Originally, {3} did not have a
symbol, and was represented by
a straight number. So 2F
was originally the only representation of {3,3,5}.
The
higher order polygons are shown in different ways. These
rarely occur
in higher-order figures, and are mainly the
provence of the new notation.
But because they can occur in
prisms, there is support for them in both
notations.
X
x/y
5/2 6/2
8
8/3 4
-------
------- ------ ------
------ ------- ------
Old
Px
PxDy
V
T
P8 P8D3 P8D2R2
New X, PX X/Y,
PX/Y 5/2
6/2
8
8/3 4
The examples for 5/2 and
6/2 are special cases, because they occur in higher
compounds
and star-polytopes: eg VFV. V and T are also used in this
meaning
in the new style also, eg oVo for o5/2o, is acceptable.
Upper and lower-cases are heavily overloaded, and the case
of an external
variable must be set into the proper case before
using it. The old style
uses lower case, the new style
uses upper case.
The D operator reflects the fact that /
refers to a marked node, and can
not also be used for a
divisor. That's what the D does. P can govern
multiple trailing D's , the purpose of the R construct is to
allow an
after-division. So PxD1S refers to
{x/1,3}. This stellates to a PxD2D1,
ie
{8/2,8/1}. But if we want to divide 8/2 through by 2, the R2
element
is used. The R notation replaces single windings
by a multiple R-fold
winding. But in PpDdRr, the actual
through-divisor used is the greatest
common divisor of (p,d,r).
The digon is "R", or "2". Supplements ie p/(p-d) are
formed with an affix
of i, so Q is 4, Qi is 4/3.
NON-REGULAR GROUPS.
The idea here is to make every
Dynkins symbol appear to be regular. This
means that we
have to deal with several problems. These examples are
real life ones from the reference collection: the hyperbolic
groups of
finite content.
A JUNCTION is a node with
three direct connections.
A LOOP is a cycle of nodes and
branches.
A TRACE is the passage through the nodes and
junctions.
In essence, we unfold the Dynkins symbol onto the
trace, and then describe
this trace. Loops are handled by
the loop-node, and junctions by higher
order subject and object
branches.
Object
: Subject
/------\
: /---------\
o o---o---o---o---o---o---o o
s o e g : C
B A S O
Old E S
S S S S
S B = ESSSSSSB = E6B
New o E o
3 o 3 o 3 o 3 o 3 o 3 o B o = oEo3o3o3o3o3o3oBo
A
Branch typically connects the subject to the object, as consectuive
nodes
in the trace. But later objects and earlier subjects
can be made use of by
the device shown here. Note that B
is presented as if it joined S to O,
but actually joins A to
O. B is a third subject, because it uses a subject
three
back. The other branches are A, C, and G.
A loop node
is a revisit after the traversing of a loop. This identifies
the current node as a revisit of the start node, or (rarely),
the second
node. If there are branches connected to the
loop, then this junction
is used as the start.
d ae 4
f
4
o----o----o
o---o---o---o---:---o
|
|
a b c d
e f
o----o
Old S S S S :
Q SSSS:Q = 4:Q
c b
New o 3 o 3 o 3 o 3 z 4 o o3o3o3o3z4o
A full tetrahedron, with all edges marked with a "3"
also occurs in the
collection. It is a subgroup of order
24 of {6,3,3}. The numbers on the
nodes refer to
verticies of the tetrahedral region. You can see here the
third-subject branches point back three on the trace, regardless
of what
is happening inside that region.
/---------\
1---2---3---4---1 1 2
\----------/
Old S S
S S : B : B :: = SSSS:B:B:: = 4:B:B::
New o 3 o 3 o 3 o 3 z B z B zz =
o3o3o3o3zBzBzz
There is occasion to have a marked branch
with a higher subject as well.
These occurs as a subgroup of
{4,4,4}. The old system supports modifiers
which greatly
expand the flexibility there. These are not used in the new
system, except that a double-branch acquires the properties of
both items.
/-----\
/--4--\
o-4-o-4-o o
o-4-o-4-o o
Old Q Q
A
Q Q Aq = QQAq
New o 4 o 4 o A o o
4 o 4 o A4 o = o4o4oA4o.
PRODUCTS
The products are formed by using a "fake" branch, ie a step
in the trace to
connect disjunct elements. The &
operator is used here. This resets the
loop count, if one
is active.
5 6
x---o---o---o * x---o {3,3,5}{6}
Old /
S S F & /
H /SSF&/H
/2F&/H
New x 3 o 3 o 3 o & x 6 o
x3o3o3o&x6o
MIRROR-MARGINS
A mirror margin is a
margin that reflects the shape of one cell into the
same shape
on the other cell. If all the margins are
mirror-margins, then
the figure is a mirror-margin, and dual to
the corresponding mirror-edge
figure. There is no native
Dynkins-node for this.
5
Dual of @---o---@---o
Old \
S S \ F \SS\F = \2\F
New m 3 o 3 m 3 o m3o3m5o
The direct product of mirror-margins is the octahedral
product. This is
the convex hull of placing the elements
in seperate orthogonal spaces with
their centres at the common
origin. Polygon Dipyramids are the line-polygon
cross.
It is for for this reason that we do not have mirror-margins
and mirror-edge
nodes in the same polytope.
Dual
of @-p-o @ p-gon
dipyramid
Old \ Pp
& \ \Pp&\
New m P o & m
mPo&m
Circles, Spheres and Cylinders
------------------------------
Just as "black is not a
colour", a "sphere is not a polytope". But just as
we can
say "x is black in colour", we can also say it is "circular in
shape".
Enclosing the circles and spheres gives these shapes
access to the products
and so forth.
The notation is as
if it were a polytope of the same dimension. O is a
letter, not zero.
Circle
{O} cf
square {4}
Sphere
{O,O} cf
cube {4,3}
4-sphere
{O,O,O} cf tesseract {4,3,3}
The "Wythoff constructs (eg /OO/ ) are used to form
ellipoids. If a node
is unmarked, it is equal to the
previous node. When / or x is used, the
axies are greater
than the before, whereas \ and m reduces the axis.
So an
oblate elipsoid (flattened sphere) has 0<x1<x2=x3, is
"/O/O".
A prolate elipsoid (stretched sphere) has
0<x1=x2<x3, is /OO/.
A cylinder is /O&/. A
Bi-circle prism is /O&/O. A pentagon-circle prism
is
x5o&xOo. And so on.
EXOTIC PRISMS
-------------
A device is used to allow for multiple
figures of the same symmetry to be
used in the same symbol,
these are "figures#for" style.
An exotic prism has a
different top and base. The notation also handles
a stack
of such prisms. Although not uniform, it is still useful,
since
the nature of the faces can be found.
The &x
suggests a prism, so #x represents an exotic prism.
5
@---@
a o o
point
Pentagonal pyramid
b x o
pentagon xb5o«
a
x o pentagon
b x
x decagon
xb5xab#xab Pentagonal cupola
a x o pentagon
b
o x pentagon
xa5xb#xab Pentagon antiprism
a x o
pentagon
Pentagon prism
b x o
pentagon
xab5o#xab top and base not nec
equal.
a o o point
b
x o pentagon
c o
x pentagon
xb5xc#xabcd (stretched) icosahderon
d o o point
The notation
of #o suggests a prism of zero height, and so the convex
hull of
the elements, with internal bits provided:
a
o x
pentagon
Decagon with 2 inscribed pentagons
b x
o pentagon
xb5xa#oab = flattened
pentagonal antiprism
The notation #c designates a cycle of
layers or a column.
a x o
pentagon
b o x
pentagon
When the 5 is replaced by an
c o x
pentagon
infinity (U), this is the alter.
d x
o pentagon
xad5xbc#cabcd triangles and square strips.
The
notation #v designates just the verticies, such as where atoms might
be situated.
o4o3o4o
a x o o o cubic
b o o o x
cubic
xa4o3o4xb#vab Body centred cubic.
o3o3o3z
a x o o {3,6}
red xa3o3o3z#vab
verticies of {6,3}
b o x o {3,6}
white xa3xb3o3z#cab Hexagonal
close-pack
c o o x {3,6}
black xa3xb3xc3#cabc Oct-tet truss.
The notation #f designates separate figures.
o3o3o3o
a x o o o
5-cell
xa3o3o3xb#fab Johnson: {(3,3,3)}
b
o o o 5 5-cell
xa3xa3xb3xb#fab Johnson: t{(3,3,3)}
This
notation can be applied to the old style notation as well.
New xabc
# So
/bF/ab#xab Pentagonal cupola
Old
/abc #
/b37/ab#xab 38d simplex Cupola
/a37/b#fab 38d "star of david"
Other
amendments
----------------
ASTERIX NODES
Asterix nodes connect the previous node to the following
node, with a branch
of 2d/d. The value of d varies from 2
to 6 as follows.
4/2 6/3
8/4 10/5 12/6
Old +
* ++ *+ **
New
<not used>
Asterix nodes are so named, because the
edges are replaced by multiple
edges crossing like an
asterix. This happens at the node where the symbol
is
applied. In fact, the two nodes that directly connect to the
asterix
nodes now become connected by a branch 4/2, 6/3, or 8/4.
The following asterix nodes have been sighted.
S+S, S+SQ, S+SQS, QS+SQ,
S+SQSS S+SQS+S
Q+Q
Q+QQ
Q+Q+Q
Pn+Pn P8+P8 vs P8++P8
H*S
T*S
Johnson calls these twinned, and writes Pp+Pp as
{(p,p)}. He offered no
notation for the nodes * or ++, or
the close + + as in "Q+Q+Q"
COMPLEX NODES
The
old style supports modifiers, and these are applied to the node
marker
"%". This carries through odd branches, and need
only be shown on the first
node of an odd set. The stary
complex figures are written, as suggested
in the form of 2p/2d,
for p/d.
One can tell such a star, since it will be of the
form "a{b/c}d", where b is
odd, and a and d are unequal. I
write this as "a{2b/2c}d", eg 3{3}2 becomes
3{6/2}2 = %P6D2.
% node marked 3
%pn node marked n
%q node marked 4
%f node marked 5
%h node marked 6
So 3{3}3{3}3 becomes %SS. 3{3}3{4}2 is %SQ,
3{3}3{4}3 is %SQ%
PETRIE NODES
------------
This
is an accomidation for the book by Coxeter on regular maps.
Coxeter
describes a series of maps, to which he writes {p,q}_r,
where r designates
the length of the petrie polygon.
Since each edge connects two verticies, faces and petrie
polygons, I write
it as if it were derived from a group:
/--r--\
o-p-o-q-o o
The node that the
petrie polygon connects to is designated as a "p", so
{3,7}_14 = x3o7oA14p
HATCH NOTATION
--------------
Don Hatch describes a class of symmetries in H^2 honeycombs,
where the
fundemental region is a polygon like a square,
pentagon or hexagon. He uses
a modified Dynkins symbol.
a---o Loop: petrie polygon of tetrahedron:
a,e cross at rightangles
| |
Polygon: real polygon = cell of {4,6} a,e do not cross.
o---e
Like the Dynkins symbol, the edges are dihedral
angles, the points are the
mirrors.
Since z can never
normally occur at the start, a leading z denotes this
polygon-cycle. There is no meaning for the direct product
in H^2, but this
notation is also valid in regular space, etc.
Hatch writes this as ( p q r s ) with | as marked
nodes, and || as snub
nodes, these trail the letters. The
symbols are cyclic. This is thence
transliteratd as
zPoQoRoSo, with x and s as required.
LAMINATE FACES
--------------
The laminate face is an unbounded
one. A laminatope is bounded by unbounded
sides. In
practice, the interior of a laminahedron is inscribed with cells,
and laminahedra are joinde together. The class example is
the alternating
squares and triangles.
In hyperbolic
space, there is a greater degree of freedom on what can go
into
a laminate face.
The aperigon is designated by "ZO".
It is featured in a figure like an
ordinary face, like "3" or
"29", except that it's infinite.
When the thing is capped,
the innards are put inside the "ZO".
Let's look at
z2x4xZs2s3sOo.
The outer figure consists of z2x4xZOo, that
is, a fairly straight-forward
t{4,oo}. It has octagons,
whose sides alternate with other octagons, and
the aperigon.
Inside this aperigon, we write "Zs2s3sO". This is yet
another normal group,
written from the interior of the polygon
ZO. Flicking this to the start, we
get zZOs2s3s.
This is just a straight snub {3,oo}. But the two aperigons
are stuck together.
In terms of Hatch's polygon, a Z..O
segment designates an internal chord or
arc that goes in at the
Z and out at the O. Entirely diverse shapes can be
on
alternate sides, as long as the aperigons match and fit together.
An aperigon is just an ordinary polygon that has an infinity
on the other
side. Use it accordingly.
The
Extended Snub
-----------------
The normal snub figure
has n edges, numbered 1 to n. Each such face is
marked eg
clockwise, and each vertex has one of each kind of vertex of the
cell at it. Inbetween each snub face, you can put any
other shape, whose
edges all marked the same number (eg "6" or
"5").
In the extended snub, the snub faces have nk edges
marked 1..n,1..n, etc.
The same rules apply, but just using "s"
is no longer suffucient.
For example, the z2s3s9s is the
regular s{3,9}, but I produced another
figure
where the "s"
segment were enneagons. This is written as z9$2s3s3s, where
the segment before the $ refers to what the "s" should be read
as.
The snub segment can be an APERIGON: ie zZO$sPsQsRs has
a meaning.
In this notation, branches marked "2" between "s"
nodes can be dropped, as
the ZO in zZO$, if there is an interior
to "ZO": eg zZss3sO$ is zss3s$...
Wendy
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