# Glossary

This glossary will define the many terms found on this site as well as many others concerning higher dimensions or polytopes.

## A

Agent - Tame simple non-orientable members of a regiment.

Arena - A complicated bowl like structure with lots of chambers that show up in various polychora - an example cross section of one is shown below.

Army - A set of polytopes with the same vertices. The convex hull is called the general.

Atypical - A uniform polytope is atypical if it is not typical. Atypicals include snubs, antiprisms, non-Wythoffian regiments like the idcossids, blends, swirlprisms, iquipadah types, and ondip types.

Axic Symmetry - The symmetry of the six dimensional cube, the hexeract - AKA Ax. symbol is ooooo'x.

Axis Product - Also called tegum product, or plus product. The axis product P of shapes A and B is the shape formed when A and B are intersecting each other at their centers in an orthogonal position and all points in between the points of A and the points of B are part of the object P. It can be written as P = A + B. The number of dimensions of P is equal to the sum of the number of dimensions of A and B. The diamond of A = A + A.

## B

Basic Element - Simplest elements of a shape. Vertices are the basic element of all polytopes, circles are the basic element of polytwisters.

Basic Shapes - Shapes that are blocks, orbs, or cross products of them. AKA circle, square, cube, cylinder, sphere, etc.

Binary Filling - A well defined filling method for all polytopes. It is the filling method most common to ray tracers. Odd density regions are solid and even density regions are hollow. The binary filling of the pentagram will leave the center part hollow.

Block - Cross product of dyads for dimensions greater than 1. Dyad for dimension 1.

Brick Symmetry - Symmetries of the blocks.

## C

Cavity - A hollow region inside a polytope or shape that is completely hidden from the outside.

Caw Filling - Short for "Core And Wedges" filling. This was my first attempt to define a filling method for rendering polychora and beyond. It takes a polyhedron (or polytope) and splits it into more simpler parts - usually into a core and a set of inflected parts (refered to as wedges). The core is filled in solid and the wedges are also solid. Anything missed is a hole. In this method all odd density regions are solid, also all non-zero density regions are solid. But there are a few zero density regions that get filled in. This filling method is not well defined for polytopes in general. See Filling Methods.

Cell - Three dimensional element of a polytope.

Central Inversion Symmetry - A symmetry that requires two opposite locations to be congruent, in other words what happens at point p happens at point -p. Examples includes the square, hexagon, cube, and dodecahedron. Counter examples includes pentagon, triangle, and tetrahedron.

Chasm - A planar shaped excavated path. Chasms will cut a 3-D object in two, but not a 4-D object.

Chosen One - Tame non-orientable member(s) of a regiment with a smaller (or smallest) number of facet types. Its facet types are either chosen ones or maximized polytopes. These can best be defined as the member(s) of the regiment with a symbol consisting of all marked nodes but with parts of the shape blending out, in other words a maximized polytope with coinciding parts blending out. All of its 2-faces (polygons) will have an even number of sides, but those of the sort x,x - x*x - and x^'x are blended out. Examples are cho, ri, garpop, graphi, and nat. Many chosen ones have choes in them.

Circle - Two dimensional orb. Solid version is sometimes called disk. Rollability of 1.

Clan - A subset of a tribe that contains one or more teepees. Their elements are of the same sort, for example if one member has decagram faces - all members will have some sort of x*'y face (x and y positive) which includes decagrams and retrodistars. For example, a Whytoffian clan could be of the sort x*y^z, where x,y, and z are all positive. If any of the variables turn negative, it would land us in another clan, but we would still be in the same tribe.

Coiloid - A group of 4-D shapes that have two rings as basic elements and its one or more facets have a continuous and uniform spiraling shape. They can be defined by two rational numbers p and q. The number of sides is equal to the least common denominator of the numerators of p and q. Convex coiloids are found amongst the dice. Their duals will have a spiral that curves back on itself as its basic element.

Colonel - Member of a polytope regiment that has a convex verf.

Complete Value - The complete value of a polytope is equal to the sum of the complete values of all of its external pieces and internal pieces, where the complete value of a dyad is counted as 1.

Combocell - When two or more cells land in the same realm and form what looks like a compound cell, it is called a "combocell". Common examples are the rhom (5-cube) shaped combocells found in many polychora from category 18, their true cells are actually cubes.

Combofacet - When two or more n dimensional facets land in the same n-space forming a compound facet, they are called combofacets. Combofaces for 2-D, combocell for 3-D elements, comboteron for 4-D, etc.

Compound - When the surface elements of a polytope (or other shape) can split into two or more polytopes (or other shapes) of the same dimensionality - it is a compound.

Contact Region - The part of a convex shape that would touch the surface of a table when in contact. It must be either one of its sides (for flat sided objects) or a section of a side. Can't be ridges, peaks, or other creased areas. Contact regions of polytopes are its sides. Contact regions of a sphere is a point, for a cylinder it is either a disk shaped face or it is a dyad (for the curved side). Curved objects have a continuum of contact regions that form each of their curved sides. Contact regions for polytwisters look like bowed out polygons.

Contic Symmetry - Symmetry of the 48-cell, aka cont. It has twice the symmetry as icoic. Symbol is ox'xo.

Cotcoic - Related to cotco or idtid. Cotcoic polychora usually have cotco or idtids for cells.

Cross Product - Also called prism product. The cross product P of shapes A and B is the shape formed by this method. Let point q be a point in A and point r be a point in B, then point {q, r} is in P. It can be written as P = A x B. The number of dimensions of P is equal to the sum of the number of dimensions of A and B. The square of A = A x A.

Cross Section - An n-1 dimensional slice of an n-dimensional polytope. Cross sections of polyhedra are polygons (or compounds of them) while cross sections of polychora are polyhedra (or compounds of them). They are like taking a CAT scan of the object and looking at one slice at a time.

Crystal - The duals of the antiprisms. Also called antitegums.

Cube (as in cube of) - The cross product of three identical shapes. The cube of a dyad is the cube itself, the cube of a square is a hexeract. The cube of a polygon is the trioprism of it. The cube of a circle is the triocylinder.

Cubinder - Cross product of square and disk, one of the prisminders. Has rollability of 1 on curved side and rollability of 0 on its four flat sides.

Cubic Symmetry - Symmetry of the cube - symbol oo'x.

Cylinder - Cross product of disk and dyad, it is the prism of a disk. Has rollability of 1 on curved side and rollability of 0 of its two flat sides.

## D

Deca Symmetry - Twice the symmetry of pennic symmetry. It is the symmetry of deca - symbol oxxo.

Defissing - Checking for and removing fissary members from a regiment.

Demi-Tessic Symmetry - Half the symmetry of the tesseract, the symmetry of sto and gotto. Symbol of symmetry is o8o.

Density Filling - A well defined filling method for orientables. All sections inside the polytope with a non-zero density is filled in, zero density areas are hollow.

Diamond - The axis product of two identical shapes. The diamond of a dyad is a square standing on its corner, the diamond of a square is a hex (16-cell). The diamond of a circle is a duospindle.

Dice - A convex shape with congruent contact regions. Important attributes are the number of sides, number of dimensions, shape of contact region, and the rollability of the dice. Two main classes are polytope dice (flat sided, contact regions are the facets) and curved.

Dircospid - Member of the gadros daskydox regiment, the most complicated uniform polychoron regiment.

Doic Symmetry - Symmetry of the dodecahedron. Symbol is oo^x.

Duoantiprism - The M-N duoantiprism can be formed by alternating the vertices of a 2M-2N duoprism.

Duoc - Refers to the octagon standing on its corner, short for dual of octagon. The duoc of an n-D shape P is formed in this manner: Take the surface of P and scale it up by a factor of sq2, make two copies and make them orthogonal - one copy in the first group of n dimensions and the other in the second group of n dimensions. Next take the cross product of the surface of P with itself, this will lay halfway between the two Ps. Connect this to the two Ps and fill it solid, this is the duoc of P, it is a powertope. The duoc of a die is also a die. The duoc of a regular polygon has wedge shaped sides. The duoc of a circle is the "Rolly Duoc".

Duocrystal - Also called duoantitegums. These are the duals of the duoantiprisms.

Duocylinder - Cross product of two disks. It can roll like a cylinder on its two congruent sides and has rollability of 1.

Duog - Refers to the octagram standing on its corner, short for dual of octagram. The duog of an n-D shape P is formed in this manner: Take the surface of P and scale it up by a factor of -sq2, make two copies and make them orthogonal - one copy in the first group of n dimensions and the other in the second group of n dimensions. Next take the cross product of the surface of P with itself, this will lay halfway between the two Ps. Connect this to the two Ps by connecting to the far side with an inflection and fill it solid, this is the duog of P, it is a powertope. The duog of a regular polygon has crossed wedge shaped sides. The duog of a circle is the "Rolly Duog".

Duogem - Also called duotegums. Axis product of two polytopes.

Duoprism - Cross product of two polytopes.

Duoring - Cross product of two circles.

Duospindle - Axis product of two disks, dual of the duocylinder. Has rollability of 2.

Dyad - Line segment. The one dimensional polytope. It has two end points.

Dyster - Short for dyadic twister. A member of the infinite set of regular polytwisters that have a number of dyad twisters as its sides.

## E

East (4-D World) - Direction of rotation.

Edge Figure - The pattern that an edge spans into. They are the verfs of the verf. Sometimes called "ef" for short.

Element - Part of a polytope, i.e. vertex, edge, face, cell, facet, ridge, peak, etc.

Even (Dice) - A die is even when a facet is always opposite to a facet.

Exotic - A "polytope" is exotic if there are any ridges that contain more than two facets. They are not considered as true polytopes.

Exotic Celled - More generically exotic faced, exotic celled, exotic teroned, or exotic faceted. They contain elements that are exotic polytopes but they themselves are not exotic. They are not true polytopes.

External Piece - A piece of a shape that is exposed to the space outside the convex hull, these pieces will need to be cut out to build a model.

## F

Face - A two dimensional element of a polytope.

Facet - The n-1 dimensional element of an n dimensional polytope (or shape).

Feral - A polytope is feral when there are cases of three elements of dimension d in the same d+1-space and meet at an element of dimension d-1 at the same time, but the d+1-space has no d+1 dimensional elements.

Field of Sections - FOS for short (fosses for plural). A two dimensional grid of poke sections, great to visualize polytera or the internal structure of a polychoron. The pic below shows an example FOS of the polychoron giphado.

Filling Method - A method used to define the interior of a polytope as well as its elements. The pic below compares various filling methods.

Fissary - A polytope like object that has peaks or lower dimensional elements that coincide completely. Fissary polychora either have compound vertex figures or edge figures that can split into two or more components.

Flune - A four dimensional space, usually in the context of being inside a higher dimension.

Funk Prism - A strange polychoron made of one kind of cell. The cell resembles a warped prism and they line up to form a trifoil knot like pattern. The warped prisms contain the following faces: 2 isosceles n-gons twisted at an angle, two kite faces joining one edge of an n-gon to one edge of the other in this pattern (n-gon---kite---kite---n-gon), two isosceles pentagons near the kites, and n-3 more kites (may have a rhombus). Convex funk prisms are amongst the gyrochoron dice and they have 3*n-2 sides. The pentagon funk prism is the tridecachoron, the square funk prism is the same as the pentagonal duoprism, the triangular funk prism is mobius 7. Higher funk prisms are called funky 16 and funky 19. When n gets large, they start to look more like the 3,2 coiloid. They sort of look like duoprisms that got mixed up with Escher's and Tim Burton's art styles. Below is the projection and the net of the 15-gon funk prism, it has 43 identical cells.

## G

Garp (4-D World) - 90 degrees in lateral rotation from north, when looking east.

Gem - Also called tegum (Wendy Krieger) or bipyramid. A gem is formed when a shape is extended in two opposite directions orthogonal to the shape to a point on both sides. It can also be called the axis product of the shape with a dyad. For example, the square gem is an octahedron.

General - The convex member of an army of polytopes.

Glome - A four dimensional orb. It is sometimes called a gongol when solid. Has rollability of 3.

Gulley - A realmic shaped excavated path. Gulleys will cut a 4-D object in two, but not a 5-D object.

Gyrochoron - Also called step tegum. Convex polychora that are bounded by a symmetric subset of the cell realms of a duogem (aka duotegum, dual of duoprism). One way to derive them is to take the vertices of an N duoprism where the vertices of the N-gon is labeled from 0 to N-1. The duoprism will have the vertices (a,b), where both a and b are from 0 to N-1. Choose N vertices of the form (a, k*a mod N) Where k is an integer greater than 1 and less than N-1. Find the dual of the convex hull of these points - this is a gyrochoron which is also a die with N sides, we could refer to it as the N-k gyro (here "-" is a dash, not a minus). When N>6, I call the N-2 gyros "Mobius N" due to the strange Mobius strip appearance of their projections. One interesting thing with gyrochora is that amongst them are dice with a prime number of sides. Some gyros look familiar, gyro 5-2 is the pentachoron, gyro 8-3 is the tesseract, gyro N-(N/2-1) is the N/2-1 duoprism, and gyro 10-3 is deca.

## H

Halforder - The number of right handed (or left handed) symmetries in a symmetry group. Doic has halforder of 60, hyic is 7200 for example.

Haxic Symmetry - Symmetry of the demihexeract, hax. Symbol - ooo8x.

Hemi Polytope - A polytope with facets going through the center and with half the number it would of had if it didn't - example: oho has 4 hexagon faces going through the center in the 3-fold locations, where normally there would be 8 faces in the 3-fold locations instead of 4.

Heppic Symmetry - Symmetry of the six dimensional simplex, the heptapeton, AKA hep - symbol is ooooox.

Hinnic Symmetry - Symmetry of the demipenteract, hin. Symbol - oo8x.

Hixic Symmetry - Symmetry of the five dimensional simplex, the hexateron, AKA hix - symbol is oooox.

Hole - A linear shaped excavated path. Holes will cut a 2-D object in two, but not a 3-D object.

Hyic Symmetry - Symmetry of the 120-cell, AKA hi or hecatonicosachorn. Symbol is ooo^x.

## I

Idcossidic - Pertaining to either the idcossids or dircospids, sometimes including the baby monster snubs.

Icoic Symmetry - Symmetry of the 24-cell, AKA ico or icosatetrachoron. Symbol is oo'ox.

Internal Piece - A piece of a shape that is not exposed to the space outside the convex hull, these pieces are exposed to hidden cavities instead.

Inter-Regimental Compound - A dyadic compound of two or more members (can be pure compound members) of the same regiment where the compound itself fits in the regiment also, an example is a compound of ico and gico (gico is a pure compound in the ico regiment) - also refered to as IR compound or IRC.

## J

Jab Section - An n-3 dimensional section of an n dimensional shape. A cross section of a cross section of a cross section. Jab sections of polypeta are 3-D.

Jakic Symmetry - Symmetry of the six dimensional polytope 221, AKA Jak - symbol oo8ox.

Jewel Regiments - Regiments of the following trend: hig - xx, co - xox, spid - xoox, scad - xooox, staf - xoooox, suph - xooooox, soxeb - xoooooox, . . . . . as well as their semi-uniform counterparts: xy, xoy, xooy, xoooy, xooooy, . . . .

## L

Laptitude (4-D World) - Measures marp and garp on a 4-D world.

Latitude (4-D World) - Measures north and south on a 4-D world.

Lone Operative - A single member regiment or its member.

Longitude (4-D World) - Measures east and west on a 4-D world.

Lieutenant - The member of a regiment that is the conjugate of the colonel in the conjugate regiment.

Lieutenant Colonel - A polytope that is both a colonel and a lieutenant.

Line - A one dimensional space, usually in the context of being inside a higher dimension.

LOC - Level of complexity. LOC = value / halforder.

Long Name - A Greek based mathematical name for a polytope, ex. "quasitruncated great stellated dodecahedron". Some polytopes have several valid long names.

## M

Mass - The mass of a region is the density of that region times the content (area, volume, bulk, etc). The mass of a shape is equal to the sum of the masses of its regions.

Marp (4-D World) - 90 degrees counter to lateral rotation from north, when looking east.

Membrane - A part of the surface of a polytope that has no solid areas on either side, this undesirable condition can happen with solid filling and CAW filling.

Moic Symmetry - Symmetry of the six dimensional polytope 122, AKA Mo - symbol oo6oo. Has twice the symmetry as jakic.

## N

Nation - This is like the "least common multiple" of tribe and regiment. Sometimes members of a tribe have regiment members in other tribes. A nation is a set of tribes who's members form regiments.

Natural Filling - Previously called neo-filling. A filling technique that is very well behaved for all polytopes. Orientable polytopes are filled by the density filling, while non-orientables are filled in a binary style. This filling method is advocated by Robert Webb (designer of Stella software) and myself. See Filling Method for more details.

Nature - The type of polytope amongst these possiblities: tame, feral, and wild. The pic shows examples.

Neo-Filling - See Natural Filling

Non Orientable - An object is non-orientable when its surface can not distinguish which way is "up". Non-orientable polytopes can not distinguish between a density of 1 and -1. I.e. Mobius strip, thah.

North (4-D World) - Surface direction opposite of the sun at highest point in the sky during winter soltice.

## O

Octagon (as in octagon of) - The octagon of an n-D shape P is formed in this manner: Let P2 be the surface of P that has been scaled up by a factor of sq2+1. Now consider these two cross products in 2n-D space: P x P2 and P2 x P. Connect these two together by connecting near sides together and fill it solid, this is the octagon of P, it is a powertope. The octagon of a regular polygon has polygon long prisms and rectangle trapezoprisms for sides. The octagon of a circle is the "Rolly Octagon".

Octagram (as in octagram of) - The octagram of an n-D shape P is formed in this manner: Let P2 be the surface of P that has been scaled down by a factor of -1/(sq2+1). Now consider these two cross products in 2n-D space: P x P2 and P2 x P. Connect these two together by connecting far sides together and fill it solid, this is the octagram of P, it is a powertope. The octagram of a regular polygon has polygon short prisms and rectangle double crossed trapezoprisms for sides and they intersect each other - the octagram of a dyad is an octagram. The octagram of a circle is the "Rolly Octagram".

Odd (Dice) - A die is odd when a facet is always opposite to a vertex.

Orb - Circle, sphere, glome, or any higher dimensional sphere. They can be either hollow or solid. Has rollability of 1 or greater.

Order - Number of symmetries in a particular symmetry group. The order of doic symmetry is 120 and the order of hyic is 14400 for example.

Orientable - An object is orientable when its surface can distinguish which way is "up" and "down". Orientable polytopes can distinguish between a density of 1 and -1, as long as the center is defined. I.e. regular polygons, sissid, gike.

## P

Peak - An n-3 dimensional element of an n-dimensional polytope. Example: vertices of polyhedra, edges of polychora, faces of polytera. If they are somewhat sharp, they will feel pointy, for example - if you were in the fourth dimension holding a tesseract, the edges (its peaks) will feel pointy (like a cube's corners).

Pennic Symmetry - Symmetry of the four dimensional simplex, the pentachoron, AKA Pen - symbol ooox.

Pentic Symmetry - Symmetry of the five dimensional cube, the pentachoron, AKA Pent - symbol oooo'x.

Pesu - Short for primary semi-uniform. These semi-uniforms are primary polytopes, but not necessarily tame.

Petsu - Short for primary tame semi-uniform. These semi-uniforms are primary and tame. There are currently 7076 known petsu polyhedron teepees, of which 6343 have the vertices of grid.

Piece - A non-intersected part of the surface of a shape that borders between empty areas and solid areas.

Pizza Graph - A polygonal graph that looks like a conglomeration of pizza slices that represents members of jewel regiments.

Plane - A two dimensional space, usually in the context of being inside a higher dimension.

Point - A zero dimensional space, usually in the context of being inside a higher dimension. It can also be a zero dimensional polytope.

Poke Section - An n-2 dimensional section of an n dimensional shape. A cross section of a cross section. Poke sections of polytera are 3-D.

Polar Equator (4-D World) - The north most equator.

Polychoron - A four dimensional polytope.

Polygon - A two dimensional polytope.

Polyhedron - A three dimensional polytope.

Polypeton - A six dimensional polytope.

Polyswirler - An eight dimensional curved polyteron like object with equatorial glomes that swirl around each other. They can roll like four dimensional spheres, no matter how they are placed on a table. They are related to quarternion Hopf fibration. They have rollability of 3.

Polytelon - A one dimensional polytope.

Polyteron - A five dimensional polytope.

Polytope - An n-dimensional shape bounded by flat facets. True polytopes must be dyadic - exactly two facets meet at each ridge (must also be true for combofacets as well) and all of its elements are true polytopes (this rules out exotics and exotic celled cases). No two of its elements can coincide completely (this rules out the fissaries). It is not a compound. Elements can however form combofacets as long as the dyadic rule fits the entire combofacet.

Polytwister - A four dimensional curved polyhedron like object with equatorial circles (rings) as the basic element that swirl around each other. Their facets are twisters and their ridges are called strips. They are related to Hopf fibration. They can roll like cylinders no matter how they are placed on a table. There are 222 uniform polytwisters with 3 infinite groups, 36 and one infinite group of which are regular. They have rollability of 1.

Powertope - A polytope (or shape) derived by taking the P of a B, which can be written as B^P. B is the base shape and P is the power. P must have some sort of brick symmetry and can have full scale n-cube symmetry and various pyritic symmetries. Higher symmetries, like doic, hyic, and icoic doesn't carry through the operation and lower symmetries are not defined. B must have a defined center (for uniforms it should be obvious). If B has central inversion symmetry, then B^P can be defined for all P, otherwise it is defined for some P. In 4-D common powers are the square, diamond, octagon, and duoc which can be defined for any shape B, with a center. The number of dimensions of a powertope is equal to the product of the number of dimensions of B and P. The following pic explains how they work.

Primary - A polytope is primary when all of its facets/combofacets line up in the directions of its symmetries (AKA no snub facets). The same goes for all of its elements according to their symmetry locations - this means that the only polygon possible in the 2-fold symmetry locations is a rectangle (including the square) - no bowties, hexagons, octagon etc in this location. 3-fold locations can have triangles (right side up or upside down only), hexagons (and ditrigons), or various tripods. For 4-fold locations, this would allow octagons (including semi-uniform), octagrams, tetrapods, square (orthogonal orientation or diamond orientation, no other), or a two rectangle comboface (plus orientation or X orientation allowed). 2-square combo (8/2 shape) isn't allowed in a 4-fold location since its edges doesn't line up to the square symmetry (orthogonal or 45 degree diagonals). There are also fissary and compound primaries. Another important thing, there can be only one primary in each location - this means that a combofacet can't split into two primaries that maintain the symmetry - for example the two rectangle combo (AKA the plus shape) will split, but each rectangle loses the 4-fold symmetry - therefore the plus shape is OK. But a combo of a square and an octagon under 4-fold symmetry is not allowed in a primary since it still maintains the 4-fold symmetry when it is split.

Prism - The extension of a shape into the next dimension, in other words, the cross product of the shape with a dyad.

Prisminder - Cross product of a circle with a polygon, they are essentially polygon-circle duoprisms. Has rollability of 1 on curved side and rollability of 0 on the flat sides.

Pseudocell - A polyhedron shaped area on a polytope that is lacking the polyhedron itself, but contains all of its faces, edges, and vertices.

Pseudoface - A polygon (including star polygon) shaped area on a polytope that is lacking the polygon itself, but contains all of its edges and vertices.

Pure Compound - A compound of only one type of polytope, like rhom or gidrissid. They may exist inside polytope regiments also.

## R

Realm - A three dimensional space, usually in the context of being inside a higher dimension.

Regiment - A set of polytopes that share the same vertices and edges.

Regiment Map - A graph structure that lists and connects all lower dimensional regiment sets that exists in a particular regiment, below is an example using the span regiment - oxoo'x.

Region - A segment of n-D space of an n-D shape that is not divided by the surface. Regions inside a star includes its 5 points and the pentagon center for example.

Regular - A shape is regular when all of its vertices (basic elements) are congruent and all of its elements of any given dimensionality is congruent and regular.

Ridge - An n-2 dimensional element of an n-dimensional polytope. Examples: edges of polyhedra, faces of polychora, cells of polytera.

Rinf - AKA. Ring figure. It is the pattern that the rings of polytwisters span into.

Ring - The basic element of a polytwister. They are equatorial circles.

Rollability - The number of dimensions that a curved object can roll freely. Cylinders and spindles have rollability of 1, a sphere has rollability of 2.

## S

Scaliform - A shape is scaliform when all of its vertices (basic elements) are congruent and all of its edges are the same length.

Section - The intersection of a shape and a lower dimensional space. They can be cross sections (1 less dimension), poke sections (2 less dimensions), jab sections (3 less dimensions), or even further for higher dimensions.

Semi-Uniform - A shape is semi-uniform when all of its vertices (basic elements for curved objects) are congruent and all of its elements are semi-uniform. Examples include rectangles, and tripods.

Sergeant - Tame non-fissary Wythoffian members (Blend of Wythoffians for blend regiments) that are not colonels nor lieutenants.

Sheet - A finite segment of a flexible plane, also called hedrix (Wendy Krieger).

Shell - The shape derived by taking the convex hull of the verf and making that the verf, most of the time it is the colonel of the regiment. Can also be called locally-convex hull.

Short Name - An abbreviated name given to a polytope that was derived from a longer mathematical name (the long name) - ex. the great ditrigonary dodekicosidodecahedron was abbreviated to GdTDID, I thus started pronouncing it as "GID ta did" and later "GID it did", the short name became "gidditdid". Some of these names are quite catchy (gogishi, wavitoth, skiviphado, padohi, getit xethi, rapsady), while others are silly sounding (picnut, frogfix, irp xanady, gun hidy, pirdy, spiddit, statupid hicadox). They have also been called Official Bowers Style Acronyms or OBSA, might as well make a short name of this - Obsa. I've had a lot of fun developing these names for the uniform polyhedra, polychora, polytera and beyond.

Simple - A polytope is simple if it is not in a blend regiment. Simple Wythoffian polytopes have verfs that have as many types of vertices as it does marked nodes.

Skew (Dice) - A die is skew when a facet is not opposite to any element.

Slice - Another name for cross section.

Slope (Dice) - A die is sloped when a facet is not opposite to any other element, however some element is nearly opposite, but is slanted in an isosceles sort of way.

Snub - A polytope with facets that are not on any main symmetry axis.

Solar Equator (4-D World) - The south most equator.

Solid Filling - A filling technique that fills every finite segment of any polytope and its elements as solid - there are no hidden cavities. This filling technique is advocated by Norman Johnson and is sometimes considered as traditional. See Filling Method for more details.

South (4-D World) - Surface direction towards the sun at highest point in the sky during winter soltice.

Sphere - A three dimensional orb, it is sometimes called a ball when solid. Has rollability of 2.

Spherinder - Cross product of a sphere and a dyad, in other words a sphere prism. Has rollability of 2 on curved side, flat side has rollability of 0.

Spitsu - Short for simple primary tame semi-uniform. There are 385 spitsu teepees under grid (has grid vertices) within 11 tribes.

Square (as in square of) - The cross product of two identical shapes. The square of a dyad is the square itself, the square of a square is a tesseract. The square of a polygon is the duoprism of it. The square of a circle is a duocylinder.

String - A finite segment of a flexible line.

Strip - The "edge" of a polytwister. It is bounded by two rings that twist around each other.

Swock - A finite segment of a flexible realm, also called chorix (Wendy Krieger).

Symbol - The symbols I use are compressed forms of the Coxeter-Dynkin diagrams which uses x's and o's and various tally marks. The following pic explains how they work:

## T

Tame - A polytope is tame when no three elements of dimension d are in the same d+1-space and meet at an element of dimension d-1 at the same time.

Tasu - Short for tame semi-uniform. They are not necessarily primary.

Teepee - A continuous set of semi-uniform polytopes (or compounds) with the same facet arrangement and the same order of vertices, edges, and other elements. Teepees can form simplex shaped areas of various morphs. The number of dimensions of the simplex is determined by the number of degrees of freedom of the morphing. Below is an example of one of the 6343 petsu teepees under grid, which has two degrees of morphing. I nicknamed the center polyhedron "the helicopter" and gave its teepee the name "pacova".

Teron - Four dimensional element of a polytope.

Tessic Symmetry - Symmetry of the four dimensional cube, the tesseract. Symbol - ooo'x.

Tettic Symmetry - Symmetry of the tetrahedron. Symbol - oox.

Traditional Filling - Same as solid filling.

Trend - A series of polytopes or regiments that increase in the number of dimensions and have similar properties. Examples include the jewel regiments, the measure polytopes, and the gocco trend (og - x"x, gocco - (o'x"x), gittith - o(o'x"x), ginnont - oo(o'x"x), goxaxog - ooo(o'x"x), gososaz - oooo(o'x"x), gook - ooooo(o'x"x), ....).

Triamond - The axis product of three identical shapes. The triamond of a dyad is an octahedron. The triamond of a circle is the 6-D triospindle.

Tribe - A continuous set of semi-uniform polytopes (or compounds) that span several teepees. Truncation rotation is an example with one variable of morphing. Wythoffian cases will include all polytopes with a particular symbol (example xy^z) allowing the variables (x,y,z, etc) to take on any value including negative. The following pic displays the grid tribe (which contains 4 clans and 60 teepees). Each teepee is displayed twice on the pic with an example polyhedron within a triangle shaped region (the sides of the dual of grid). The grid tribe's clans are xy^z (grid), x,y^z (reboga), xy^'z (badori), and x,y^'z (robisu) - where x,y,and z take on positive values. There are 11 spitsu tribes under grid, they are xy^z (grid, 60 teepees), x*y^z (quitdid, 60 teepees), xy*z (gaquatid, 60 teepees), (x^y*'z) (idtid, 60 teepees), (xy*z) (becada, 30 teepees), (x^y^z,) (fabeca, 30 teepees), (xy^'z) (cafeta, 30 teepees), (x*y*z,) (mocaba, 30 teepees), (x^'y^'z^') (jefari, 10 teepees), (x*y*z*) (vamesa, 10 teepees), and the 15-block compound tribe x y z (broza, 5 teepees). I suspect that four dimensional tribes, such as the gidpixhi tribe could have as many as 7200 teepees with three variables of morphing.

Troop - Member of a regiment.

Truncation Rotation - A continuum cycle of truncations. Starting at the base polytope at 12-o-clock, truncating it until the rectate is reached at 3-o-clock. Continue to the hypertruncations until 6-o-clock. Keep going through the quasitruncations until 9-o-clock. Then finally through the inflected cases back to 12. See pic below:

Typical - A uniform polytope is typical if it is in a regiment described by a Coxeter-Dynkin diagram with no alternation nodes (AKA no snubs) and its symmetry is represented by a potential CD diagram of the colonel (excludes swirlprisms and iquipadah like members). AKA they are Wythoffian symmetric members of Wythoffian regiments. Example: Sishi and sidpith are Wythoffian regiments, members such as gaghi, paphacki, stefacoth, and snipto are typical, but the swirlprisms and iquipadah are not. Rico can have the following as symbols: oo'xo xox'o x6x so any member with icoic, tessic, or demitessic symmetry inside this regiment are typical.

Twister - The facet of a polytwister. They look like bloated polygonal rods that are twisted 360 degrees and curved into a cycle.

## U

Uniform - A shape is uniform when all of its vertices (basic elements for curved objects) are congruent, its elements are uniform, and its edges are the same length.

## V

Value - The value of a polytope is equal to the sum of the complete values of all of its external pieces, where the value of a dyad is counted as 1.

Verf - AKA vertex figure. This is the pattern that the vertices spread into. Example: the verf of a cube is a triangle.

Verf Structure - A complete verf for a particular regiment, or a projection of it, that shows lines for all possible polygons inside the regiment. All equivalent classes of polygons are usually displayed as color coded lines. Drawing the verf structure of all possible lower regiment classes inside the regiment can show how these regiments fit inside the verf.

## W

Web - An n-2 or lower dimensional part of the surface of an n-D polytope that has no solid areas touching it, this undesirable condition can happen with solid filling and CAW filling.

Wedge (Dice) - A die is wedge when a facet is opposite to an element other than a facet or vertex.

West (4-D World) - Direction against rotation.

Wild - A polytope is wild when there are cases of three elements of dimension d in the same d+1-space and meet at an element of dimension d-1 at the same time, and the d+1-space has d+1 dimensional elements.

Winding Method - A method to find the density of regions within an orientable polytope.

Wint - A direction in four dimensional space. It is 90 degrees clockwise from left.

## Z

Zant - A direction in four dimensional space. It is 90 degrees clockwise from right.

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