Site Map  Polytopes  Dynkin Diagrams  Vertex Figures, etc.  Incidence Matrices  Index 
The notion of all these different products was mainly inspired by Miss W. Krieger.
The setup of the ,>,O devices originates independently to Mr. P. Pugeau and Quickfur.
The pyramid product within the mostoften used sense describes the stepup to the following dimension, using any former polytope as a base, and adding a single vertex in an orthogonal direction atop that base as a tip, thereby getting as new polytope that spanned pyramid.
For any given subdimension the subelements of that new polytope consist out of those subelements of that very dimension of the base polytope, plus the alike derived pyramids with the same tip but using any of the subelements of the base polytope with one dimension less.  This extends through all subdimension, including that of the full space on the one end ("body"), and including the empty space ("nulloid") at the other. (The nulloid, or empty set, of any polytope is unique, has dimension 1, and is incident to all subelements, including the vertices. It is the dual of the polytopal body.) The product of the empty space at the base and a single vertex (tip) being that very vertex.
The pyramid product applies especially to the dimensional set of simplices (sometimes also called: the series of pyropolytopes). Here we get the Pascal triangle:
elemental  encasing dimension counts  1 0 1 2 3 4 5 6 7 8 9 10  ({3}) (tet) (pen) (hix) (hop) (oca) (ene) (day) (ux) + 1  1  1  1  1  1  1  1  1  1  1  1  1  \ \ \ \ \ \ \ \ \ \ \ sub 0  1  2  3  4  5  6  7  8  9  10  11 elemen  \ \ \ \ \ \ \ \ \ \ tal 1  1  3  6  10  15  21  28  36  45  55 dimen  \ \ \ \ \ \ \ \ \ sion 2  1  4  10  20  35  56  84  120  165  \ \ \ \ \ \ \ \ 3  1  5  15  35  70  126  210  330  \ \ \ \ \ \ \ 4  1  6  21  56  126  252  462  \ \ \ \ \ \ 5  1  7  28  84  210  462  \ \ \ \ \ 6  1  8  36  120  330  \ \ \ \ 7  1  9  45  165  \ \ \ 8  1  10  55  \ \ 9  1  11  \ 10  1
This small table can also be expressed as coefficients of the rational function: (1/x) · (x+1)^{D+1}, where D is the encasing dimension and the actual power of x denotes the subelemental dimension.
The circumradius of the simplex can be given as a function of its dimension too, just as its inradius – and thus its height, and even the dihedral angle, or its volume:
circumradius = sqrt(D)/sqrt[2(D+1)] inradius = 1/sqrt[2D(D+1)] height = sqrt(D+1)/sqrt(2(D+d)) in orient. acc. to d(D1d) volume = sqrt[(D+1)/(2^{D})]/D! surface = (D+1) sqrt[D/(2^{D1})]/(D1)! dihedral angle = arccos(1/D)
(Esp. for D → ∞ we get for the regular simplex circumradius = 0(D1)height = 1/sqrt(2), inradius = volume = 0, and dihedral angle = 90°.)
Beyond this narrower sense we also could build a pyramid product of any 2 nondegenerate polytopes. Then this product will position either factor in an orthogonal subspace, but shift those subspaces relative to oneanother, along a direction mutually orthogonal to both. Then all will be subject to a convex hull – at least as long both factors are convex themselves. Accordingly the dimension of this product will be the sum of the dimensions of the factors, plus one. For nonconvex factors those bases could be morphed similarily, by dimensional degression, running over the set of lacing facets, which surely are pyramid products of one dimension less.
The neutral element of that product clearly is the nulloid. The pyramid product in the narrower sense is just the restriction hereof to one factor being a mere point. Further, in case both factors are uniform simplices, and the shift is adjusted so that the lacing edges also have unit lengths, this product results in (higher dimensional) uniform simplices again.
The general notation here is ×_{1,1}, i.e. it shall consider both, the (unique) nulloid and the (unique) bulk. Let P and Q be D_{P} resp. D_{Q}dimensional polytopes, represented by their respective sets of (sub)elements P = {n_{d,i} P_{d,i} : d dimension ∈ [1,D_{P}], i type ∈ I_{d}}, where the coefficients n_{d,i} represent the absolute counts (according to the diagonal elements of the corresponding incidence matrix) and P_{d,i} represent all types of (sub)elemental polytopes; and Q similar. Then this operator can be defined in its bivalent form by the segmentopal atop notation as:
P ×_{1,1} Q = = {n_{d,i} P_{d,i} : d dimension ∈ [1,D_{P}], i type ∈ I_{d}} ×_{1,1} {m_{e,j} Q_{e,j} : e dimension ∈ [1,D_{Q}], j type ∈ J_{e}} = {n_{d,i} · m_{e,j} (P_{d,i}  perp Q_{e,j}) : d ∈ [1,D_{P}], e ∈ [1,D_{Q}], i ∈ I_{d}, j ∈ J_{e}} = P  perp Q e.g. triangle ×_{1,1} line = = {1 nulloid, 3 vertices, 3 lines, 1 triangle} ×_{1,1} {1 nulloid, 2 vertices, 1 line} = {1 (nulloid  perp nulloid), 3 (vertex  perp nulloid), 3 (line  perp nulloid), 1 (triangle  perp nulloid), 2 (nulloid  perp vertex), 6 (vertex  perp vertex), 6 (line  perp vertex), 2 (triangle  perp vertex), 1 (nulloid  perp line), 3 (vertex  perp line), 3 (line  perp line), 1 (triangle  perp line)} = {(1+0+0) nulloid, (3+2+0) vertices, (3+6+1) lines, (1+6+3) triangles, (0+2+3) tet, (0+0+1) pen} = pen
Note that the perp part in the atop notation here is quite relevant, as without that the atop notation e.g. iterates in its multivalent form into an axial stack like a tower only, while the pyramid product clearly becomes a true associative product, where any pairing contributes like in general simplices.
A bit more general than the simplices obtained from the pyramid product of the former section are the simplicial polytopes, i.e. all those polytopes, which have simplices for facets only. In fact, the here obtained convex polytopes are being enlisted on the page on deltahedra (which there includes their higher dimensional counterparts too). As then the types of faces of each dimension will be unique (at least by shape – not so by aroundsymmetry in general), the according numbers can be nonumbiguously be given as f_{k} for 1 ≤ k ≤ d (with f_{1} = f_{d} = 1).
For those simplicial polytopes then the set of DehnSummervill equations can be stated, i.e. for 1 ≤ k ≤ d2 the following equations do hold:
d1 Σ
j=k 
(1)^{j} 
(

j+1 k+1 
)

f_{j} = (1)^{d1} f_{k} 
More explicitely for simplicial polyhedra there holds the following set of equations:
f_{1} + f_{0}  f_{1} + f_{2} = f_{1} _{ } f_{0}  2f_{1} + 3f_{2} = f_{0} _{ } _{ } f_{1} + 3f_{2} = f_{1} _{ } _{ } _{ } f_{2} = f_{2}
respectively for simplicial polychora one has the following set of equations:
f_{1} + f_{0}  f_{1} + f_{2}  f_{3} = f_{1} _{ } f_{0}  2f_{1} + 3f_{2}  4f_{3} = f_{0} _{ } _{ } f_{1} + 3f_{2}  6f_{3} = f_{1} _{ } _{ } _{ } f_{2}  4f_{3} = f_{2} _{ } _{ } _{ } _{ } f_{3} = f_{3}
or for simplicial polytera one gets the following set of equations:
f_{1} + f_{0}  f_{1} + f_{2}  f_{3} + f_{4} = f_{1} _{ } f_{0}  2f_{1} + 3f_{2}  4f_{3} + 5f_{4} = f_{0} _{ } _{ } f_{1} + 3f_{2}  6f_{3} + 10f_{4} = f_{1} _{ } _{ } _{ } f_{2}  4f_{3} + 10f_{4} = f_{2} _{ } _{ } _{ } _{ } f_{3} + 5f_{4} = f_{3} _{ } _{ } _{ } _{ } _{ } f_{4} = f_{4}
and so on.
The prism product essentially is that product elsewhere in geometry described as cartesian crossproduct, or sometimes even as direct sum. If both components have encasing dimensions larger than one, the result will be the duoprism thus derived. If just one component is a mere edge (1 dimensional), the product will build up the orthogonal prism atop the other component (base), and if both are 1 dimensional we get the square. Clearly, the 0 dimensional polytopal element (i.e. a point) is the neutral element of this product. Further we could consider the product of n factors, accordingly this would lead to a multiprism (triprism, quadprism, etc.).
If we just consider the stepup by 1 dimension, i.e. the second factor being a mere orthogonal edge, a similar dimensional increasing table can be derived: The subdimensional elements of any order are either those of the 2 opposite bases (doubling that number), or the prisms atop those subelements of the bases, which have 1 dimension less (adding that count).  This applies for all subdimensions of the bases, including the body, but excluding the nulloid!
The prism product applies especially to the dimensional set of hypercubes (sometimes also called: the series of geopolytopes):
elemental  encasing dimension counts  0 1 2 3 4 5 6 7 8 9 10  ({4}) (cube) (tes) (pent) (ax) (hept) (octo) (enne) (deker) + sub 0  1 = 2 = 4 = 8 = 16 = 32 = 64 = 128 = 256 = 512 = 1024 elemen  \ \ \ \ \ \ \ \ \ \ tal 1  1 = 4 = 12 = 32 = 80 = 192 = 448 = 1024 = 2304 = 5120 dimen  \ \ \ \ \ \ \ \ \ sion 2  1 = 6 = 24 = 80 = 240 = 672 = 1792 = 4608 = 11520  \ \ \ \ \ \ \ \ 3  1 = 8 = 40 = 160 = 560 = 1792 = 5376 = 15360  \ \ \ \ \ \ \ 4  1 = 10 = 60 = 280 = 1120 = 4032 = 13440  \ \ \ \ \ \ 5  1 = 12 = 84 = 448 = 2016 = 8064  \ \ \ \ \ 6  1 = 14 = 112 = 672 = 3360  \ \ \ \ 7  1 = 16 = 144 = 960  \ \ \ 8  1 = 18 = 180  \ \ 9  1 = 20  \ 10  1
This small table can also be expressed as coefficients of the polynomial: (x+2)^{D}, where D is the encasing dimension and the actual power of x denotes the subelemental dimension.
The circumradius of the hypercube can be given as a function of its dimension too, just as its inradius, and even the angle α between adjacent facet normals, or its volume:
circumradius = sqrt(D)/2 inradius = 1/2 volume = 1 surface = 2D dihedral angle = 90°
(Esp. for D → ∞ the circumradius too becomes ∞.)
The general notation here is ×_{0,1}, i.e. it shall not consider the (unique) nulloid, but still the (unique) bulk. Let P and Q be once more D_{P} resp. D_{Q}dimensional polytopes, represented by their respective sets of (sub)elements P = {n_{d,i} P_{d,i} : d dimension ∈ [1,D_{P}], i type ∈ I_{d}}, where the coefficients n_{d,i} represent the absolute counts (according to the diagonal elements of the corresponding incidence matrix) and P_{d,i} represent all types of (sub)elemental polytopes; and Q similar. Then this bivalent operator is defined by:
P ×_{0,1} Q = = {n_{d,i} P_{d,i} : d dimension ∈ [1,D_{P}], i type ∈ I_{d}} ×_{0,1} {m_{e,j} Q_{e,j} : e dimension ∈ [1,D_{Q}], j type ∈ J_{e}} = {1 nulloid} ∪ {n_{d,i} m_{e,j} (P_{d,i} × Q_{e,j}) : d ∈ [0,D_{P}], e ∈ [0,D_{Q}], i ∈ I_{d}, j ∈ J_{e}} e.g. triangle ×_{0,1} pentagon = = {1 nulloid, 3 vertices, 3 lines, 1 triangle} ×_{0,1} {1 nulloid, 5 vertices, 5 lines, 1 pentagon} = [1 nulloid} ∪ {15 (vertex × vertex), 15 (line × vertex), 5 (triangle × vertex), 15 (vertex × line), 15 (line × line), 5 (triangle × line), 3 (vertex × pentagon), 3 (line × pentagon), 1 (triangle × pentagon)} = {1 nulloid, (15+0+0) vertices, (15+15+0) lines, (5+0+0) triangles, (0+15+0) squares, (0+0+3) pentagons, (0+5+0) trip, (0+0+3) pip, (0+0+1) trapedip} = trapedip
Wrt. the circumradius one here generally has the theorem of Pythagoras, for sure: r^{2}(P ×_{0,1} Q) = r^{2}(P) + r^{2}(Q).
Tegum derives from latin and is ment in the sense of a coating skin of a tent. In fact, as long only convex shapes as factors are considered, the tegum product positions those within orthogonal (nonafine) subspaces, and covers the whole by its convex hull. Accordingly the dimension of the product is just the sum of the dimensions of the factors.
Esp., if one factor is just 1 dimensional, the tegum product becomes the dipyramid with equatorial crosssection being the other factor. In that case the subdimensional counts derive either from those of the crosssection of the corresponding dimension, or from those of the 2 pyramid products based each on the subdimensional elements of the crosssection of one dimension less. Considering just the counts, here those 2 pyramid products and a tegum product of that very base with an edge, would get the same numbers. This can be used for a similar dimensional iteration as for the other cases. – Again that rule applies for all subdimensional elements, here including the nulloid (which is the neutral element of the product here), but excluding the body (as the crosssection itself is not a facet of the dipyramid).
The tegum product applies especially to the dimensional set of orthoplexes (a.k.a. crosspolytopes – others call it the series of aeropolytopes):
elemental  encasing dimension counts  0 1 2 3 4 5 6 7 8 9 10  (qedge)({4}) (oct) (hex) (tac) (gee) (zee) (ek) (vee) (ka) + 1  1  1  1  1  1  1  1  1  1  1  1  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ sub 0  2  4  6  8  10  12  14  16  18  20 elemen  \\ \\ \\ \\ \\ \\ \\ \\ \\ tal 1  4 12  24  40  60  84  112  144  180 dimen  \\ \\ \\ \\ \\ \\ \\ \\ sion 2  8  32  80  160  280  448  672  960  \\ \\ \\ \\ \\ \\ \\ 3  16  80  240  560  1120  2016  3360  \\ \\ \\ \\ \\ \\ 4  32  192  672  1792  4032  8064  \\ \\ \\ \\ \\ 5  64  448  1792  5376  13440  \\ \\ \\ \\ 6  128  1024  4608  15360  \\ \\ \\ 7  256  2304  11520  \\ \\ 8  512  5120  \\ 9  1024
This small table can also be expressed as coefficients of the rational function: (1/x) · (2x+1)^{D}, where D is the encasing dimension and the actual power of x denotes the subelemental dimension.
The circumradius of the orthoplexes can be given as a function of its dimension too, just as its inradius, and even the angle α between adjacent facet normals, or its volume:
circumradius = sqrt(1/2) inradius = sqrt(1/2D) volume = sqrt[2^{D}]/D! surface = 2 sqrt[2^{D1} D]/(D1)! dihedral angle = arccos(2/D  1)
(Esp. for D → ∞ we get inradius = volume = 0 and dihedral angle = 180°, which shows that the ∞dimensional orthoplex becomes a flat honeycomb, in fact one with a finite circumradius!)
As a nice coincidence consider the general formula: cos(2a) = 2 cos^{2}(a)  1. Applying that to the dihedral angle of the regular simplex provides: cos(2 arccos(1/D)) = 2 cos^{2}(arccos(1/D))  1 = 2/(D^{2})  1. I.e. the dihedral angle of the D^{2} dimensional orthoplex equates to twice the angle of the D dimensional simplex!
The general notation here is ×_{1,0}, i.e. it shall consider the (unique) nulloid, but not the (unique) bulk. Let P and Q be once more D_{P} resp. D_{Q}dimensional polytopes, represented by their respective sets of (sub)elements P = {n_{d,i} P_{d,i} : d dimension ∈ [1,D_{P}], i type ∈ I_{d}}, where the coefficients n_{d,i} represent the absolute counts (according to the diagonal elements of the corresponding incidence matrix) and P_{d,i} represent all types of (sub)elemental polytopes; and Q similar. Then this bivalent operator (restricted to convex figures) can be defined using the hull and the orthogonal sum operator (A ⊕ B = (A,0) ∪ (0,B)) by:
P ×_{1,0} Q = = {n_{d,i} P_{d,i} : d dimension ∈ [1,D_{P}], i type ∈ I_{d}} ×_{1,0} {m_{e,j} Q_{e,j} : e dimension ∈ [1,D_{Q}], j type ∈ J_{e}} = {n_{d,i} m_{e,j} hull(P_{d,i} ⊕ Q_{e,j}) : d ∈ [1,D_{P}1], e ∈ [1,D_{Q}1], i ∈ I_{d}, j ∈ J_{e}} ∪ {1 hull(P ⊕ Q)} e.g. triangle ×_{1,0} pentagon = = {1 nulloid, 3 vertices, 3 lines, 1 triangle} ×_{1,0} {1 nulloid, 5 vertices, 5 lines, 1 pentagon} = {1 hull(nulloid ⊕ nulloid), 3 hull(vertex ⊕ nulloid), 3 hull(line ⊕ nulloid), 5 hull(nulloid ⊕ vertex), 15 hull(vertex ⊕ vertex), 15 hull(line ⊕ vertex), 5 hull(nulloid ⊕ line), 15 hull(vertex ⊕ line), 15 hull(line ⊕ line)} ∪ {1 hull(triangle ⊕ pentagon)} = {(1+0+0) nulloid, (3+5+0) vertices, (3+0+5) Sedges, (0+15+0) Ledges, (0+15+15) SLLtriangles, (0+0+15) So2oS&#L} ∪ {1 hull(triangle ⊕ pentagon)} = hull(triangle ⊕ pentagon)
(Short edges (S) of the provided example are those of the defining unitedged triangle resp. pentagon. Long edges then are of size sqrt(r_{{3}}^{2}+r_{{5}}^{2}) = sqrt[(25+3 sqrt(5))/30] = 1.056940, connecting the respective vertices of the defining perpendicular polygons.)
In general the tegum product is dual to the prism product. But as the Catalans generally are not unitedged, and as can be seen from the afore mentioned example, esp. the length of the lacing edges highly depends on the shape of the defining factors. E.g. when one factor is a qedge and the other is a unitsized orthoplex, then the product becomes the unitsized orthoplex in the next dimension, i.e. it happens to be all unitedged again, as the above table shows. Despite the nonunit sized qedge (which is nothing but the exceptional 1D member of unit orthoplexes) one even gets the more general result, that the tegum product of a ddimensional unit orthoplex and a Ddimensional unit orthoplex happens to be nothing but the (d+D)dimensional unit orthoplex. – But there is a prominent other such all unitedged example, resulting from unitedged factors, the pentagon ×_{1,0} pentagram.
Having mentioned the tegum product, being the hull of the fully orthogonal sum, i.e. A ×_{1,0} B = hull( A ⊕ B ) = hull((A,0) ∪ (0,B)), we get a close relation to the mere tegum sum, which is defined by A ◊ B = hull( A + B ) = hull( A ∪ B ). Within the extended Dynkin symbol notations this tegum sum translates into that of degenerate zeroheight lace prisms with pseudo bases:
extendeddyn(P ◊ Q) = 2layerdyn(P, Q) &#zx e.g. extendeddyn( x,wrectangle ◊ w,xrectangle ) = xw2wx&#zx ( = x4x, octagon) extendeddyn( qtet ◊ dual qtet ) = qo3oo3oq&#zx ( = x4o3o, cube)
For sure, the tegum sum of 2 fully orthogonally placed components again results in the tegum product of those components.
Note that this A ∪ B itself will be nothing but the compound of A and B. Thus the tegum sum of the addends (or "layers") alternatively could be called the hull of the compound of the components.
This product again is a cartesian crossproduct or direct sum. Therefore nulloids do not contribute in the sequential hierarchy. But because tilings and honeycombs are infinite polytopes without body, that one is to be omitted here as well. Further, as total counts are infinite (and within this infinitude even with an exponent according to the filled dimension), only relative frequences make sense. In fact, vertex counts are to be multiplied (the = or \\marked "additions" really should be represented by Ntuple lines here, and N → ∞); this results in prefactors for vertex counts (resp. the relative frequences) which all equal 1.
The honeycomb product applies especially to the dimensional set of hypercubical honeycombs.
elemental  filled dimension counts  0^{ } 1^{ } 2^{ } 3^{ } 4^{ } 5^{ } 6^{ } 7^{ } 8^{ } 9^{ } 10 (N → ∞)  (aze)^{ } (squat)^{ } (chon)^{ } (test)^{ } (penth)^{ } (axh)^{ } + 0  N^{0} = N^{1} = N^{2} = N^{3} = N^{4} = N^{5} = N^{6} = N^{7} = N^{8} = N^{9} = N^{10}  ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ sub 1  ^{ } N^{1} = 2.N^{2} = 3.N^{3} = 4.N^{4} = 5.N^{5} = 6.N^{6} = 7.N^{7} = 8.N^{8} = 9.N^{9} = 10.N^{10} elemen  ^{ } ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ tal 2  ^{ } ^{ } N^{2} = 3.N^{3} = 6.N^{4} = 10.N^{5} = 15.N^{6} = 21.N^{7} = 28.N^{8} = 36.N^{9} = 45.N^{10} dimen  ^{ } ^{ } ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ sion 3  ^{ } ^{ } ^{ } N^{3} = 4.N^{4} = 10.N^{5} = 20.N^{6} = 35.N^{7} = 56.N^{8} = 84.N^{9} = 120.N^{10}  ^{ } ^{ } ^{ } ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ 4  ^{ } ^{ } ^{ } ^{ } N^{4} = 5.N^{5} = 15.N^{6} = 35.N^{7} = 70.N^{8} = 126.N^{9} = 210.N^{10}  ^{ } ^{ } ^{ } ^{ } ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ 5  ^{ } ^{ } ^{ } ^{ } ^{ } N^{5} = 6.N^{6} = 21.N^{7} = 56.N^{8} = 126.N^{9} = 252.N^{10}  ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ 6  ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } N^{6} = 7.N^{7} = 28.N^{8} = 84.N^{9} = 210.N^{10}  ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } \\ ^{ } \\ ^{ } \\ ^{ } \\ 7  ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } N^{7} = 8.N^{8} = 36.N^{9} = 120.N^{10}  ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } \\ ^{ } \\ ^{ } \\ 8  ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } N^{8} = 9.N^{9} = 45.N^{10}  ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } \\ ^{ } \\ 9  ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } N^{9} = 10.N^{10}  ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } \\ 10  ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } ^{ } N^{10}
This small table can also be expressed as coefficients of the polynomial: [N · (x+1)]^{D}, where D is the filled dimension and the actual power of x denotes the subelemental dimension.
The general notation here is ×_{0,0}, i.e. it shall neither consider the (unique) nulloid, nor the (unique) bulk. (Note that within this context we speak of infinite polytopes, not of tesselations. When considering tesselations, the not to be used bulk surely is omitted already.) Let P and Q be once more D_{P} resp. D_{Q}dimensional polytopes, represented by their respective sets of (sub)elements P = {n_{d,i} P_{d,i} : d dimension ∈ [1,D_{P}], i type ∈ I_{d}}, where the coefficients n_{d,i} represent the absolute counts (according to the diagonal elements of the corresponding incidence matrix) and P_{d,i} represent all types of (sub)elemental polytopes; and Q similar. Then this bivalent operator can be defined by:
P ×_{0,0} Q = = {n_{d,i} P_{d,i} : d dimension ∈ [1,D_{P}], i type ∈ I_{d}} ×_{0,0} {m_{e,j} Q_{e,j} : e dimension ∈ [1,D_{Q}], j type ∈ J_{e}} = [1 nulloid} ∪ {n_{d,i} m_{e,j} (P_{d,i} × Q_{e,j}) : d ∈ [0,D_{P}1], e ∈ [0,D_{Q}1], i ∈ I_{d}, j ∈ J_{e}} ∪ {1 (P × Q)} e.g. trat ×_{0,0} aze = = lim_{ N → ∞} {1 nulloid, N vertices, 3N edges, 2N triangles, 1 bulk} ×_{0,0} lim_{ M → ∞} {1 nulloid, M vertices, M lines, 1 bulk} = {1 nulloid} ∪ lim_{ N,M → ∞} {NM (vertex × vertex), NM (vertex × line), 3NM (line × vertex), 3NM (line × line), 2NM (triangle × vertex), 2NM (triangle × line)} ∪ {1 (trat × aze} = lim_{ K → ∞} {1 nulloid, (K+0+0) vertices, (K+3K+0) lines, (0+3K+0) squares, (0+0+2K) triangles, (0+0+2K) trips, 1 bulk} = tiph
This section serves as a short intro to the ,>,O devices (provided independently by P. Pugeau and Quickfur), its relations to the affore mentioned products, and then esppecially will be devoted to explicite applications.
These devices contain 3 symbols. Each represents a specific building operation, all increasing the dimension by 1. They all start with a mere 1D element, the edge, being displayed by . (In fact, one might start with a point . instead. But then, the action of any of the symbols, being apllied to it, would result in a line segment. Therefore that part can be omitted, and the number of symbols directly corresponds to the dimension of the resulting figure.)
The first such operation is the extrusion (""). It is nothing but the afore mentioned prism product.
It uses some object as base, and extrudes that object up to the opposite base, therby nowhere changing that crosssection. This operation is
likewise represented by  (sometimes also alternatively by the more typewriterready capital letter "I"),
being attached at the right to the so far compiled ,>,O device for the base.
Especially  therefore describes the square.
–
Algebraically the application of .. could be described by abs(...  a_{n} x_{n}) + abs(... + a_{n} x_{n}) = a_{0}.
The second operation is the tapering (">") of some base with one opponent point.
It thus describes the afore mentioned pyramid product.
Its symbol is a > sign (sometimes also alternatively by the more typewriterready capital letter "A"),
attached to the right of the so far compiled ,>,O device for the base.
The simplest example here is >, describing the triangle.
–
Algebraically the application of ..> could be described by abs(...  a_{n} x_{n}) + abs(...) = a_{0}.
Finally this symbolic set also includes a spin ("O") advice, producing all sorts of round things.
The action of that operation will be denoted by a trailing O symbol.
(Note, that this is the point where Quickfurs notion would differ. He originally just intended some round bitapering, which led to lots
of crude shapes. The spin operation of Pugeau here definitely serves better, providing plenty of highly interesting shapes.)
–
Algebraically the application of the latter ..O could be described by sqrt((...)^{2} + a_{n}^{2} x_{n}^{2}) = a_{0}.
But as we here are related to polytopes, we restrict here to few examples only:
Whereas the spin operation in effect seems to commute with either one of the formers, extrusion and tapering definitely do not. – In the followings we will omit any applications of the spin. Restricted thus, then any iterated application of both, of extrusion and of tapering, clearly remains within the set of segmentotopes only. This is because each such application just adds a single additional vertex layer within a further, perpendicular dimension.
As usual, we then will consider unit edges only. Surely this does not provide any restriction to the extrusions. But for the taperings this amounts for the starting figure (corresponding to the very symbol, but without that last > sign) that those should have an circumradius which is lower than unity: else lacing edges of unit size cannot connect the base vertices to the tip anymore! (Note that the circumradius itself here clearly does exist, as we are dealing with segmentotopes.)
1D  2D  3D  4D  5D  6D 

  edge 
  {4} >  {3} 
  cube >  squippy >  trip >>  tet 
  tes (R = 1) >  cubpy (R = 1) >  squippyp >>  squasc >  tisdip >>  trippy >>  tepe >>>  pen 
  pent (R > 1) >  tespy (H = 0) >  cubpyp (R > 1) >>  cubasc (H = 0) >  squasquippy (R = 1) >>  squippyippy (R = 1) >>  squascop >>>  squete >  tracube (R > 1) >>  tisdippy (R > 1) >>  trippyp >>>  trippasc >>  squatet >>>  tepepy >>>  penp >>>>  hix 
  ax (R > 1) >  (pentpy) † >  (tespyp) ° (R > 1) >>  (tesasc) ‡ >  squacubpy (R > 1) >>  (cubpyippy) † >>  (cubascop) ° (R > 1) >>>  (cubete) ‡ >  cusquippy (R > 1) >>  squasquippypy (H = 0) >>  squippyippyp (R > 1) >>>  squippypasc (H = 0) >>  squisquasc (R = 1) >>>  squascoppy >>>  squetep >>>>  squepe >  tratess (R > 1) >>  (tracubpy) † >>  tisdippyp (R > 1) >>>  (tisdipasc) † >>  squatrippy (R > 1) >>>  trippyippy >>>  tripascop >>>>  tripete >>  tetcube (R > 1) >>>  squatetpy >>>  tepepyp >>>>  tepasc >>>  squapen >>>>  penppy >>>>  hixip >>>>>  hop 
When considering the volume formulae provided above, it often might serve not only interesting when having the absolute values, but likewise those providing the ratios with respect to the corresponding circum and insphere.
Speaking of sphere or hypersphere within this context, we mean not its surface, but rather the union with its bulk underneath, sometimes also called the hyperball. I.e. in 2D we won't speak of the circle, but rather of the disk. – Further, as the volume of the Dsphere scales according to Dth power of its radius, we reserve the term "unit sphere" for those with unit radius (not diameter!).
The problem in here is that the volume of the to be used Dsphere cannot be given in a closed form, except when recruiting to the nonintuitive Gamma function. So we have either to distinguish between even and odd dimensions, or we could refuge to general comparisions of dimensions D+2 with D.
volume of unit Dsphere = π^{D/2} / Γ(D/2 + 1) volume of unit 2Dsphere = π^{D} / [Π_{k=1}^{D} k] = π^{D} / D! volume of unit 2D+1sphere = π^{D} / [Π_{k=1}^{D+1} (k  1/2)] = 2 D!(4π)^{D} / (2D+1)! volume ratio of unit Dsphere with unit (D2)sphere = 2π / D starting values for recursion (and few further values): volume of unit 0sphere = 1 volume of unit 1sphere = 2 volume of unit 2sphere = π^{ } = 3.141593 volume of unit 3sphere = 4 π / 3^{ } = 4.188790 volume of unit 4sphere = π^{2} / 2 = 4.934802 volume of unit 5sphere = 8 π^{2} / 15 = 5.263789 volume of unit 6sphere = π^{3} / 6 = 5.167713 volume of unit 7sphere = 16 π^{3} / 105 = 4.724766 volume of unit 8sphere = π^{4} / 24 = 4.058712 ... volume of unit ∞sphere = 0
(That last value already shows the reason for that section: we are measuring volumes in general by hypercubical units. And we will see, that the higher the dimension, the more relative volumina will be centered at the vertex surroundings of the hypercube! I.e. the volume outside its insphere, compared to the insphere volume, gets arbitrarily large.)
For simplices we get for every even dimension that the ratio of the simplex volume V_{simplex} to the volume of the respective insphere V_{insphere} evaluates to
V_{simplex}(D) : V_{insphere}(D) = sqrt[(D+1)/(2^{D})]/D! : ( π^{D}/D! · 1/sqrt[2D(D+1)]^{D} ) = sqrt(D+1)[sqrt(D(D+1))/π]^{D}, if D = 2n
For D → ∞ this value increases without bound. That is, the simplex eventually will have nearly all its volume outside of its insphere!
For orthoplexes we get for every even dimension that the ratio of the orthoplex volume V_{orthoplex} to the volume of the respective insphere V_{insphere} evaluates to
V_{orthoplex}(D) : V_{insphere}(D) = sqrt(2^{D})/D! : ( π^{D}/D! · 1/sqrt(2D)^{D} ) = [2sqrt(D)/π]^{D}, if D = 2n
For D → ∞ this value increases without bound, but to a lesser degree. That is, the orthoplex still eventually will have nearly all its volume outside of its insphere!
For hypercubes we get for every even dimension that the ratio of the hypercube volume V_{hypercube} to the volume of the respective insphere V_{insphere} evaluates to
V_{hypercube}(D) : V_{insphere}(D) = 1 : ( π^{D}/D! · 1/2^{D} ) = (2/π)^{D} D!, if D = 2n
For D → ∞ this value again increases without bound (in fact being for every allowed D just within the bounds of the corresponding values of simplex resp. orthoplex). That is, also the hypercube eventually will have nearly all its volume outside of its insphere!
When comparing various polytopes one might ask which one is more like a hypersphere than the other. Above we already gave a first answer to that by the comparison of the volume of the polytope to the volume of its insphere. This serves well for regular polytopes. But in general there will be no common insphere. One might want to dualize that very setup, comparing the volume of the circumsphere to the volume of the polytope. But again this might serve for orbiform polytopes only. Thus we have to dive a bit deeper for an even more general concept.
Such a general objective measure of roundness can be provided by the ratio of its body volume to its surface volume. But this sort of measure still depends on the absolute size because body volume and surface volume do scale differently wrt. their absolute size variable. For that reason let us consider the following absolute roundness measure
abs.roundness(P) = (V_{P})^{dim(∂P)} / (V_{∂P})^{dim(P)}
where V_{P} is the volume of some polytope P, ∂P is its (hyper)surface, and dim(P) is its (local) dimension. Esp. dim(∂P) = dim(P)1 for sure. By the use of these exponents this measure now becomes scale independent. It is also known as the isoperimetric ratio.
If we further denote by abs.roundness(O) the absolute roundness of the (filled hyper)sphere itself, then one sees a further disadventage of this so far defined measure (cf. the according explicitely provided values below): the absolute roundness is good for comparing polytopes of the same dimension, but for comparing shapes of various dimensions, e.g. of the different hyperspheres themselves, the according values vary highly with their dimensions. – This is why we further introduce the relative roundness measure as the ratio
rel.roundness(P) = abs.roundness(P) / abs.roundness(O)
Thus, just by definition, one gets rel.roundness(O) = 1 for the hyperspheres O of all dimensions. Therefore the dimensional sequences of according values for classes of polytopes P now become comparable as well.
The volume of the Dsphere, V_{O}, was already given above. The surface contents of the Dsphere, V_{∂O}, can be derived via differentiation wrt. to its radius R therefrom as
V_{∂O}(R) = dV_{O}(R) / dR surface of unit Dsphere = D π^{D/2} / Γ(D/2 + 1) = D · volume of unit Dsphere surface of unit 2Dsphere = 2 π^{D} / (D1)! surface of unit 2D+1sphere = 2 D!(4π)^{D} / (2D)!
Thus for the here relevant values of abs.roundness(O) one gets therefrom the following general formula for even resp. odd dimensions. Additionally some values for the lower dimensions are provided explicitely.
abs.roundness( 2Dsphere ) = D! / (π^{D} (2D)^{2D}) abs.roundness( 2D+1sphere ) = (2D)! / (π^{D} (2D+1)^{2D} 2^{2D+1} D!) abs.roundness( 1sphere ) = 1 / 2^{ }^{ }^{ } = 0.5 abs.roundness( 2sphere ) = 1 / (2^{2} π)^{ }^{ } = 0.079577 abs.roundness( 3sphere ) = 1 / (2^{2} 3^{2} π)^{ } = 0.0088419 abs.roundness( 4sphere ) = 1 / (2^{7} π^{2})^{ } = 0.00079157 abs.roundness( 5sphere ) = 3 / (2^{3} 5^{4} π^{2}) = 0.000060793 abs.roundness( 6sphere ) = 1 / (2^{5} 3^{5} π^{3}) = 0.0000041476 abs.roundness( 7sphere ) = 3·5 / (2^{4} 7^{6} π^{3}) = 0.00000025700 abs.roundness( 8sphere ) = 3 / (2^{21} π^{4})^{} = 0.000000014686 ...
Having done those calculations for the hypersphere, we turn to the series of simplices now. Their volumes and surface contents were already given above.
abs.roundness( Dsimplex ) = D! / sqrt[D^{3D}(D+1)^{D+1}] abs.roundness( line ) = 1 / 2^{ }^{ }^{ } = 0.5 abs.roundness( {3} ) = sqrt(3) / (2^{2} 3^{2})^{ } = 0.048113 abs.roundness( tet ) = sqrt(3) / (2^{3} 3^{4})^{ } = 0.0026729 abs.roundness( pen ) = 3 sqrt(5) / (2^{9} 5^{3})^{ } = 0.00010482 abs.roundness( hix ) = sqrt(5) / (3^{2} 5^{7})^{ } = 0.0000031802 abs.roundness( hop ) = 5 sqrt(7) / (2^{5} 3^{7} 7^{4 }) = 0.000000078728 abs.roundness( oca ) = 3^{2} 5 sqrt(7) / (2^{8} 7^{10}) = 0.0000000016464 abs.roundness( ene ) = 5·7 / (2^{29} 3^{7})^{ } = 0.000000000029809 ... rel.roundness( 2Dsimplex ) = (2D)! π^{D} / (D! sqrt[(2D)^{2D}(2D+1)^{2D+1}]) rel.roundness( 2D+1simplex ) = D! (2π)^{D} / ((D+1)^{D+1} sqrt[(2D+1)^{2D+1}]) rel.roundness( line ) = 1^{ }^{ }^{ }^{ } = 100 % rel.roundness( {3} ) = π sqrt(3) / 3^{2}^{ }^{ }^{ } = 60.459979 % rel.roundness( tet ) = π sqrt(3) / (2 3^{2})^{ }^{ }^{ } = 30.229989 % rel.roundness( pen ) = 3 π^{2} sqrt(5) / (2^{2} 5^{3})^{ } = 13.241464 % rel.roundness( hix ) = 2^{3} π^{2} sqrt(5) / (3^{3} 5^{3}) = 5.231196 % rel.roundness( hop ) = 5 π^{3} sqrt(7) / (3^{2} 7^{4})^{ } = 1.898165 % rel.roundness( oca ) = 3 π^{3} sqrt(7) / (2^{4} 7^{4})^{ } = 0.640631 % rel.roundness( ene ) = 5·7 π^{4} / (2^{8} 3^{8})^{ } = 0.202982 % ...
Next we consider the hypercubes. Again the volumes and surface contents are given above. (Because of being measure polytopes, those here happen to be rather trivial for sure.)
abs.roundness( Dhypercube ) = 1 / (2D)^{D} abs.roundness( line ) = 1 / 2^{ }^{ }^{ } = 0.5 abs.roundness( {4} ) = 1 / 2^{4}^{ }^{ } = 0.0625 abs.roundness( cube ) = 1 / (2^{3} 3^{3})^{ } = 0.0046296 abs.roundness( tes ) = 1 / 2^{12}^{ } = 0.00024414 abs.roundness( pent ) = 1 / (2^{5} 5^{5})^{ } = 0.00001 abs.roundness( ax ) = 1 / (2^{12} 3^{6}) = 0.00000033490 abs.roundness( hept ) = 1 / (2^{7} 7^{7})^{ } = 0.0000000094865 abs.roundness( octo ) = 1 / 2^{32}^{ } = 0.00000000023283 ... rel.roundness( 2Dhypercube ) = (π/4)^{D} / D! rel.roundness( 2D+1hypercube ) = π^{D} D! / (2D+1)! rel.roundness( line ) = 1^{ }^{ }^{ } = 100 % rel.roundness( {4} ) = π / 2^{2}^{ }^{ } = 78.539816 % rel.roundness( cube ) = π / (2·3)^{ }^{ }^{ } = 52.359878 % rel.roundness( tes ) = π^{2} / 2^{5}^{ } = 30.842514 % rel.roundness( pent ) = π^{2} / (2^{2} 3·5)^{ } = 16.449341 % rel.roundness( ax ) = π^{3} / (2^{7} 3)^{ } = 8.074551 % rel.roundness( hept ) = π^{3} / (2^{3} 3·5·7)^{ } = 3.691223 % rel.roundness( octo ) = π^{4} / (2^{11} 3) = 1.585434 % ...
Then there is the series of orthoplexes (aka crosspolytopes). Their volumes and surface contents are provided above as well.
abs.roundness( Dorthoplex ) = D! / (2^{D} sqrt(D^{3D})) abs.roundness( line ) = 1 / 2^{ }^{ }^{ } = 0.5 abs.roundness( {4} ) = 1 / 2^{4}^{ }^{ } = 0.0625 abs.roundness( oct ) = sqrt(3) / (2^{2} 3^{4})^{ } = 0.0053458 abs.roundness( hex ) = 3 / 2^{13}^{ }^{ } = 0.00036621 abs.roundness( tac ) = 3 sqrt(5) / (2^{2} 5^{7})^{ } = 0.000021466 abs.roundness( gee ) = 5 / (2^{11} 3^{7})^{ } = 0.0000011163 abs.roundness( zee ) = 3^{2} 5 sqrt(7) / (2^{3} 7^{10}) = 0.000000052686 abs.roundness( ek ) = 3^{2} 5·7 / 2^{37}^{ } = 0.0000000022919 ... rel.roundness( 2Dorthoplex ) = π^{D} (2D)! / (2^{3D} D^{D} D!) rel.roundness( 2D+1orthoplex ) = π^{D} D! / sqrt[(2D+1)^{2D+1}] rel.roundness( line ) = 1^{ }^{ }^{ }^{ } = 100 % rel.roundness( {4} ) = π / 2^{2}^{ }^{ }^{ } = 78.539816 % rel.roundness( oct ) = π sqrt(3) / 3^{2}^{ }^{ }^{ } = 60.459979 % rel.roundness( hex ) = 3 π^{2} / 2^{6}^{ }^{ } = 46.263771 % rel.roundness( tac ) = 2 π^{2} sqrt(5) / 5^{3}^{ }^{ } = 35.310570 % rel.roundness( gee ) = 5 π^{3} / (2^{6} 3^{2})^{ } = 26.915171 % rel.roundness( zee ) = 2·3 π^{3} sqrt(7) / 7^{4}^{ }^{ }= 20.500183 % rel.roundness( ek ) = 3·5·7 π^{4} / 2^{16}^{ }^{ } = 15.606620 % ...
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