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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 most-often used sense describes the step-up 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 pyro-polytopes). 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||(D-1-d) volume = sqrt[(D+1)/(2^{D})]/D! dihedral angle = arccos(1/D)
(Esp. for D → ∞ we get for the regular simplex circumradius = 0||(D-1)-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 non-degenerate polytopes. Then this product will position either factor in an orthogonal subspace, but shift those subspaces relative to one-another, 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 non-convex 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 here-of 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.
The prism product essentially is that product elsewhere in geometry described as cartesian cross-product, 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 step-up 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 geo-polytopes):
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 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 (non-afine) 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 cross-section being the other factor. In that case the subdimensional counts derive either from those of the cross-section of the corresponding dimension, or from those of the 2 pyramid products based each on the subdimensional elements of the cross-section 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 cross-section itself is not a facet of the dipyramid).
The tegum product applies especially to the dimensional set of orthoplexes (a.k.a. cross-polytopes – others call it the series of aero-polytopes):
elemental | encasing dimension counts | 0 1 2 3 4 5 6 7 8 9 10 | ({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! 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) S-edges, (0+15+0) L-edges, (0+15+15) SLL-triangles, (0+0+15) So2oS&#L} ∪ {1 hull(triangle ⊕ pentagon)} = hull(triangle ⊕ pentagon)
(Short edges (S) of the provided example are those of the defining unit-edged 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 unit-edged, 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 q-edge and the other is a unit-sized orthoplex, then the product becomes the unit-sized orthoplex in the next dimension, i.e. all unit-edged again, as the above table shows. But there is a prominent other such all unit-edged example, 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 zero-height lace prisms with pseudo bases:
extended-dyn(P ◊ Q) = 2-layer-dyn(P, Q) &#zx e.g. extended-dyn( x,w-rectangle ◊ w,x-rectangle ) = xw2wx&#zx ( = x4x, octagon) extended-dyn( q-tet ◊ dual q-tet ) = 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 cross-product 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 N-tuple 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 cross-section. This operation is
likewise represented by | (sometimes also alternatively by the more typewriter-ready 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 typewriter-ready 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 bi-tapering, 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 ||>>| - squascop ||>>> - squete |>||| - tracube (R > 1) |>||> - tisdippy |>|>| - 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 ||>|>> - squippypasc ||>>|| - squasquasc (R = 1) ||>>|> - squascoppy ||>>>| - squetep ||>>>> - squepe |>|||| - tratess (R > 1) |>|||> - (tracubpy) † |>||>| - tisdippyp |>||>> - tisdipasc |>|>|| - squatrippy (R > 1) |>|>|> - trippyippy |>|>>| - tripascop |>|>>> - tripete |>>||| - tetcube (R > 1) |>>||> - squatetpy |>>|>| - tepepyp |>>|>> - tepasc |>>>|| - squapen |>>>|> - penppy |>>>>| - hixip |>>>>> - hop |
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