WOLFRAM NOTEBOOK

WOLFRAM|DEMONSTRATIONS PROJECT

Zonotope Construction via Shephard's Theorem

​
dimension
2D
3D
data
graphics
raw data
displayed 2D tiles
16
displayed 3D tiles
10
permutation number
3
Given
X={
a
1
,…,
a
r
}
, a set of generators for
n
, the zonotope
(X)
generated by
X
is the convex hull of all vectors of the form
∑
a∈X
a
,; that is,
(X)
is the Minkowski sum of all segments
[0,a]
, where
a∈X
. Also,
(X)
is the shadow of the
r
-dimensional cube
r
[0,1]
via the projection
(
t
1
,…,
t
r
)↦
r
∑
i=1
t
i
a
i
.
Given any
S⊂X
such that
S
is a basis of
n
and
λ∈
n
, the Minkowski sum
λ
(S)=λ+(S)
is called a parallelepiped. The zonotope
(X)
is said to be paved by a collection
Π
of parallelepipeds whenever
(X)=
⋃
∈Π
and
⋂
is at most
(n-1)
-dimensional for
≠∈Π
.
Shephard's theorem explicitly provides a recursively generated collection of vectors
Λ={
λ
i
}
such that
λ
i
(
S
i
)
S
i
⊂X,
S
i
abasisof
n
is a paving of
(X)
. Each tile of this paving (i.e. each parallelepiped of the form
λ
i
(
S
i
)=
λ
i
+(
S
i
)
) is shown in the figure in a different color.
Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.