Tegum product

Tegum product
The 3D pentagonal bipyramid is the tegum product of a 2D pentagon and a 1D line segment (outlined in red).
Symbol	${\displaystyle \oplus }$
Rank formula	${\displaystyle m+n}$
Element formula	${\displaystyle (m-1)\times (n-1)+1}$
Dual	Prism product
Algebraic properties
Algebraic structure	Commutative monoid
Associative	Yes
Commutative	Yes
Identity	Point
Annihilator	Nullitope
Uniquely factorizable	Yes

The tegum product, direct sum or free sum is an operation that can be applied on any two input polytopes. It extends the notion of a bipyramid (also called tegum), where one of the input polytopes is a dyad. The simplest tegum products that cannot be represented as simple tegums are the polygonal duotegums in 4D.

The tegum product of two polytopes can be called a duotegum or duopyramid, although the latter term can also refer to the pyramid product. The tegum product of n  polytopes can be called an n -tegum. In the general case, these are called multitegums.

Tegum products are particularly applicable to orthoplexes. The tegum product of an m -orthoplex and an n -orthoplex is an (m +n )-orthoplex. In particular, an n -orthoplex can be seen as the tegum product of n  dyads.

It is one of the four fundamental products on abstract polytopes, the other three being the pyramid product (join), prism product (Cartesian product), and comb product (topological product). All four of these operations are commutative, associative, and uniquely factorizable with the exception of the annihilator. The identity of the tegum product is the point, and the annihilator is the nullitope. Tegum products are closely related to pyramid products, but are differentiated mainly by the additional dimension required by pyramid products.

The tegum product was originally created for convex polytopes but also applies to abstract polytopes and their realizations. The term tegum originates from the hi.gher.space community and is used particularly in writing on polytopes interesting due to symmetry.

Definition

For convex polytopes

The direct sum ${\displaystyle P\oplus Q}$ of two convex polytopes ${\displaystyle P\subseteq \mathbb {R} ^{n}}$ and ${\displaystyle Q\subseteq \mathbb {R} ^{m}}$ is defined as the convex hull of all points of the form ${\displaystyle (p,0)}$ where ${\displaystyle p\in V(P)}$ and all points of the form ${\displaystyle (0,q)}$ where ${\displaystyle q\in V(Q)}$. Here ${\displaystyle V(P)}$ is the vertex locations of P , and the notation ${\displaystyle (p,0)}$ means appending the coordinates of vector p  with zeros until it is a vector in ${\displaystyle \mathbb {R} ^{n+m}}$.

Put another way, the direct sum places the two polytopes in orthogonal subspaces of ${\displaystyle \mathbb {R} ^{n+m}}$ and takes the convex hull of them.

On geometrical polytopes, the direct sum is sensitive to translation, and requires a specified origin. In the context of polytopes with a well-defined center, it is typically assumed that they are centered on the origin.

For abstract polytopes

The tegum product was generalized to abstract polytopes by Gleason and Hubard. The tegum product of two abstract polytopes defined by posets P  and Q  is the direct product of P  and Q , with all of the elements (p,q) where exactly one of p and q is of maximal rank taken out. In other words, this is the poset on

${\displaystyle \{(p,q):p\in P{\text{ and }}q\in Q{\text{ and either none or both of }}p{\text{ and }}q{\text{ are maximal}}\},}$

with the order relation such that

${\displaystyle (p_{1},q_{1})\leq (p_{2},q_{2}){\text{ iff }}p_{1}\leq p_{2}{\text{ and }}q_{1}\leq q_{2}.}$

This can be contrasted with the pyramid product, which does not omit any elements from the direct product, or with the prism product, which omits those with elements of minimal rank instead.

For abstract polytopes with realizations

Given proper elements ${\displaystyle p\in P}$ and ${\displaystyle q\in Q}$, the rank in ${\displaystyle P\oplus Q}$ turns out to be

${\displaystyle {\text{rank}}_{P\oplus Q}(p,q)={\text{rank}}_{P}(p)+{\text{rank}}_{Q}(q)+1}$

Therefore, the vertices of ${\displaystyle P\oplus Q}$ are either of the form ${\displaystyle ({\text{nullitope}},{\text{vertex}})}$ or ${\displaystyle ({\text{vertex}},{\text{nullitope}})}$. This leads naturally to vertex locations in the realization of ${\displaystyle P\oplus Q}$, which are formed by adding zeros to the end of vertex coordinate locations in P and prepending them to vertex coordinate locations in Q.

Properties

The tegum product is "dual" to the prism product:

${\displaystyle \delta (A\oplus B)=\delta (A)\times \delta (B)}$

where the Greek letter delta represents the dual operation.

Inherited from base polytopes

A tegum product is:

• isotopic if both bases are isotopic
  • uniform dual if both bases are uniform dual
• isogonal if both bases are isogonal and congruent to each other
• noble if both bases are noble and congruent to each other

In general, a duotegum cannot be adjusted to have regular faces, except for some bipyramids and other rare instances such as the pentagonal-pentagrammic duotegum. Specifically, the tegum product of regular-faced polytopes where both are at least 2-dimensional with circumradii ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ can be made with regular faces if and only if ${\displaystyle r_{1}^{2}+r_{2}^{2}=1}$. Non-trivial scaliform examples include the 6-dimensional great snub dodecicosidodecahedral duotegum and the 8-dimensional hollow small stellated dodecahedral antiprismatic duotegum.

Rank

The rank of the tegum product of two polytopes A and B is equal to the sum of their ranks:

${\displaystyle {\text{rank}}(A\oplus B)={\text{rank}}(A)+{\text{rank}}(B)}$

As a result, up to 3D, tegum products correspond to bipyramids. The simplest non-degenerate duotegum that cannot be represented as a bipyramid is the 4D triangular duotegum.

Inradius

The inradius of the prism product of a set of polytopes equals the inverse square root of the sum of the inverse squared inradii of each of the polytopes:

${\displaystyle {\text{inrad}}(A\oplus B)={\frac {1}{\sqrt {{\text{inrad}}(A)^{-2}+{\text{inrad}}(B)^{-2}}}}}$

This is based on the inverse Pythagorean theorem.

Facets and vertex figures

The facets of a tegum product are the pyramid products of the facets of both polytopes:

${\displaystyle F(A\oplus B)=F(A)\bowtie F(B)}$

where F(X)  is the set of all facets of polytope X .

The vertex figures of the tegum product of two polytopes will be the tegum products of the vertex figures of each polytope with the other polytope:

${\displaystyle V(A\oplus B)=V(A)\oplus B{\text{ and }}A\oplus V(B)}$

where V(X)  is the set of all vertex figures of polytope X .

Element counts

The element counts by rank may be computed as follows. If ${\displaystyle p_{r}}$ is the number of elements of P of rank r, and ${\displaystyle q_{r}}$ is the number of elements of Q of rank r, then the convolution of the sequences ${\displaystyle p_{0},p_{1},\ldots ,p_{n}}$ and ${\displaystyle q_{0},q_{1},\ldots ,q_{m}}$ produces the equivalent sequence for ${\displaystyle P\oplus Q}$.

Examples

The pentagonal bipyramid is the tegum product of a pentagon and a line segment. Bipyramids in general are formed by the tegum product of a polygon and a line segment.

The n-orthoplex is the tegum product of n congruent line segments.

External links

• Klitzing, Richard. "The Tegum Product"