# Direct sum

(Redirected from Tegum product)
Direct sum
The 3D pentagonal bipyramid is the direct sum of a 2D pentagon and a 1D line segment (outlined in red).
Symbol${\displaystyle \oplus }$[1]
Rank formula${\displaystyle n+m}$
Element formula${\displaystyle (n-1)\times (m-1)+1}$
DualPrism product
Algebraic properties
Algebraic structureCommutative monoid
AssociativeYes
CommutativeYes
IdentityPoint
AnnihilatorNullitope
Uniquely factorizableYes[note 1][1]

The direct sum[1] or free sum[2] is an operation that can be applied on any two base polytopes, generalizing the formation of bipyramids from polygons. It takes a rank-n and a rank-m polytope as operands and produces a polytope of rank n + m. It was originally created for convex polytopes but also applies to abstract polytopes and their realizations.

Despite the name, it is classified as a polytope product and 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). It is also called the tegum product, a term originating from the hi.gher.space community and used particularly in writing on polytopes interesting due to symmetry. The direct sum of two polytopes is sometimes called a duotegum, and polytopes resulting from direct sums on n polytopes can be called n-tegums or multitegums.

## 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.

### For abstract polytopes

The direct sum was generalized to abstract polytopes by Gleason and Hubard.[1] Let P and Q be abstract polytopes as partially ordered sets. The direct sum of the polytopes is defined as the direct product of P and Q (in the binary relation sense), 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 relation such that

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

### 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 direct sum is commutative and associative. Gleason and Hubard proved that the direct sum admits a unique prime factorization theorem for all polytopes except the nullitope. The nullitope is the annihilator; the direct sum of any polytope with the nullitope is the nullitope. If a polytope cannot be expressed as a direct sum of two polytopes, it is called prime with respect to the direct sum.

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.

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 ${\displaystyle P\oplus Q}$.

### Relations to other polytope products

The direct sum is closely related to the prism product ${\displaystyle \times }$. For abstract polytopes we have

${\displaystyle P\oplus Q=(P^{*}\times Q^{*})^{*}}$

where ${\displaystyle P^{*}}$ denotes the dual. For convex polytopes, if both polytopes contain the origin, we have the similar identity ${\displaystyle P\oplus Q=(P^{\circ }\times Q^{\circ })^{\circ }}$ where ${\displaystyle ^{\circ }}$ is the polar set.

The facets of a direct sum are the pyramid products of the facets of one polytope and the entirety of the other polytope.

### For specific types of polytopes

The direct sum of a vertex-transitive polytope with itself is also vertex-transitive, and the direct sum of a facet-transitive polytope with itself is also facet-transitive. Consequently, the direct sum of two congruent noble polytopes is noble.

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 direct 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 ${\displaystyle r_{1}^{2}+r_{2}^{2}=1}$.

## Examples

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

The n-orthoplex is the direct sum of n congruent line segments.

## Notes

1. With the exception of the annihilator.

## References

1. Gleason, Ian; Hubard, Isabel (2018). "Products of abstract polytopes" (PDF). Journal of Combinatorial Theory, Series A. 157: 287–320. doi:10.1016/j.jcta.2018.02.002.
2. Kalai, Gil (1989). "The number of faces of centrally-symmetric polytopes". 5 (1): 389-391. doi:10.1007/bf01788696. Cite journal requires |journal= (help)