### Different Products, occuring with Polytopes

This content was mainly inspired by Miss W. Krieger.

#### The Pyramid Product   (up)

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
|                        (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
```

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 angle α between adjacent facet normals, or its volume:

```circumradius = sqrt[D/2(D+1)]
height       = sqrt[(D+1)/2D]
cos(α)       = -1/D
volume       = sqrt[(D+1)/2^D]/D!
```

(Esp. for D → ∞ we get for the regular simplex  circumradius = height = 1/sqrt(2),   inradius = volume = cos(α) = 0.)

Beyond this narrower sense we also could build a pyramid product of 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. E.g. for 3D we have: point || 3gon, line || perp line, and 3gon || point, all describing nothing but the tet.

#### The Prism Product   (up)

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
|                  (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
```

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
cos(α)       = 0
volume       = 1
```

(Esp. for D → ∞ the circumradius too becomes .)

#### The Tegum Product   (up)

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
|                   (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
```

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)
cos(α)       = 1-2/D
volume       = sqrt[2^D]/D!
```

(Esp. for D → ∞ we get inradius = volume = α = 0, the latter of which shows that the -dimensional orthoplex becomes a flat honeycomb, in fact one with a finite circumradius!)

#### The Honeycomb Product   (up)

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)
-------------+--------------------------------------------------------------------------------------------------
0  |  N0  =  N1  =    N2  =    N3  =    N4  =     N5  =     N6  =     N7  =     N8  =      N9  =      N10
|      \\     \\       \\       \\       \\        \\        \\        \\        \\         \\
sub-      1  |         N1  =  2.N2  =  3.N3  =  4.N4  =   5.N5  =   6.N6  =   7.N7  =   8.N8  =    9.N9  =   10.N10
elemen-      |             \\       \\       \\       \\        \\        \\        \\        \\         \\
tal       2  |                  N2  =  3.N3  =  6.N4  =  10.N5  =  15.N6  =  21.N7  =  28.N8  =   36.N9  =   45.N10
dimen-       |                      \\       \\       \\        \\        \\        \\        \\         \\
sion      3  |                           N3  =  4.N4  =  10.N5  =  20.N6  =  35.N7  =  56.N8  =   84.N9  =  120.N10
|                               \\       \\        \\        \\        \\        \\         \\
4  |                                    N4  =   5.N5  =  15.N6  =  35.N7  =  70.N8  =  126.N9  =  210.N10
|                                        \\        \\        \\        \\        \\         \\
5  |                                              N5  =   6.N6  =  21.N7  =  56.N8  =  126.N9  =  252.N10
|                                                  \\        \\        \\        \\         \\
6  |                                                        N6  =   7.N7  =  28.N8  =   84.N9  =  210.N10
|                                                            \\        \\        \\         \\
7  |                                                                  N7  =   8.N8  =   36.N9  =  120.N10
|                                                                      \\        \\         \\
8  |                                                                            N8  =    9.N9  =   45.N10
|                                                                                \\         \\
9  |                                                                                       N9  =   10.N10
|                                                                                           \\
10  |                                                                                                  N10
```