Law of Sums of the Squares of Areas,
Volumes and Hyper Volumes of Regular
Polytopes from Clifford Algebras
Carlos Castro Perelman∗, Fang Fang, Klee Irwin
Quantum Gravity Research Group, Topanga, California 90290, USA
November 2012
Abstract
Inspired by the recent sums of the squares law obtained by [1] we derive
the law of the sums of the squares of the areas, volumes and hyper-volumes
associated with the faces, cells and hyper-cells of regular polytopes in
diverse dimensions after using Clifford algebraic methods [3].
1
Introduction
The sums of squares law derived by [1] states that the ratio of the sums of
the squares of the edge-lengths of a regular polytope, before and after their
orthogonal projection from D to lower D′-dimensions, is given by the ratio of
dimensions D′/D. Such sums of squares law was a direct consequence of the
Schur orthogonality formula and the edge-transitive property of the point group
G that allows to break the order of the group |G| into an integer multiple of the
number edges |G| = kE, and to recast any inner product of edges < eb, ea >
in the form < Rb(ea), ea >, where Rb(ea) is the linear map (represented as a
D ×D matrix) which sends edge ea into edge eb.
The Schur orthogonality relations [2] (see Appendix) among the matrix com-
ponents R
(g)
nm and associated with the real-valued D-dimensional irreducible rep-
resentation of the point group G are given by
|G|∑
g
R(g)
nm R
(g)
n′m′ = δnn′ δmm′
|G|
dim V
= δnn′ δmm′
|G|
D
(1a)
where one has dim V = D. In particular
|G|∑
g
R
(g)
n1 R
(g)
n′1 = δnn′ δ11
|G|
D
(1b)
∗perelmanc@hotmail.com
1
where one sums over the number of group elements g denoted by |G|. If one
takes the trace over the n, n′ indices but now with the restriction that n, n′
range from 1 to D′ ( since we are projecting down to the subspace S of D′-dim
), and if instead of summing over all the group elements from 1 to |G|, one sums
only over the E group elements associated with one representative of each one
of the respective E cosets gH corresponding to the (normal) subgroup H of G
that leaves fixed a chosen edge, then one gets in the right hand side of (1b)
n=D′∑
n=1
E∑
1
Rn1 Rn1 =
E D′
D
(2)
If the lengths La = 1 of all the edges of the regular polytope (polyhedra) in
D-dim are normalized to unity, for convenience , then
∑E
a=1(La)
2 = E, hence
from eq-(2), after dividing both sides by E =
∑E
a=1(La)
2, one arrives at
1
∑E
a=1(La)
2
E∑
a=1
n=D′∑
n=1
(R
(a1)
n1 )
2 =
1
∑E
a=1(La)
2
E∑
a=1
(L′a)
2 =
D′
D
(3)
which is the sum of squares law described by [1] in a nutshell.
This can be seen by choosing the reference edge ~e(1) to point along the
direction of the first coordinate axis such that the D-components of this fiducial
edge are ~e(1) = (1, 0, 0, ......., 0) ⇒ e(1)
1 = 1. If R
(a1)(e(1)) represents the linear
map (matrix) which sends edge ~e(1) to edge ~e(a), after projecting R(a1)(~e(1))
onto the subspace S of D′-dimensions, and spanned by the unit basis vectors
~un, where the values of n are now restricted to be n = 1, 2, 3, ....D
′, one finds
that the latter matrix R(a1)has for non-vanishing components the following
R
(a1)
n1 ones.
Therefore, the square of the orthogonal projection of each edge of unit length
onto the subspace S of D′-dimensions is given by
(L′a)
2 =
n=D′∑
n=1
(< ~e(a), ~un >)
2 =
n=D′∑
n=1
(< R(a1)~e(1), ~un >)
2 =
n=D′∑
n=1
(< (R
(a1)
n1 e
(1)
1 ) ~un, ~un >)
2 =
n=D′∑
n=1
(R
(a1)
n1 )
2
(4)
hence, after inserting eq-(4) into the left hand side of eq-(3) it yields the sum of
length squares law, see [1] for further details.
One can generalize the sums of length squares result of [1] to the sums of
areas, volumes, hyper-volumes squared by recurring to generalized transforma-
tions involving bi-vector area coordinates xµ1µ2 = −xµ2µ1 ; tri-vector volume co-
ordinates xµ1µ2µ3 = −xµ2µ1µ3 , ...., and poly-vector coordinates xµ1µ2...µp . The
index p ranges from 1 to D. The generalized coordinates in Clifford spaces [3] are
associated with a Clifford-valued X = XMΓM coordinate in D-dimensions and
2
corresponding to a Clifford algebra Cl(D). For example, in D = 4 dimensions
one has that X can be expanded in a Clifford basis as
X = s 1 + xµ γµ + x
µν γµ∧γν + xµνρ γµ∧γν∧γρ + xµνρτ γµ∧γν∧γρ∧γτ (5)
It is very natural to construct group representations of face-transitive, cell-
transitive, hyper-cell-transitive groups in terms of generalized matrices (not
to be confused with hyper-matrices) with polyvector-valued indices. And as
such, one can extend the Schur orthogonality relations to generalized matri-
ces. The Schur orthogonality relation among the generalized matrix compo-
nents R
(g)
[n1n2] [m1m2]
involving bi-vector valued indices and associated with the
real-valued D(D−1)
2
-dimensional irreducible representation of the face-transitive
group G′ , which maps any face of the regular polytope into another congruent
face, is given by a generalization of the formula (1b)
|G′|∑
g
R
(g)
[n1n2] [12]
R
(g)
[n′1n
′
2] [12]
= δ[n1n2] [n′1n′2] δ[12] [12]
|G′|
D(D − 1)/2
(6)
If instead of edge-transitivity one assumes now face-transitivity, thenR
(a1)
[n1n2] [12]
are the generalized matrices entries corresponding to the generalized transfor-
mations (poly-rotations, for example) which sends the fiducial face f (1), pointing
along the bivector basis direction e[12] = e1 ∧ e2 = γ1 ∧ γ2, onto any other face
f (a) (another bi-vector) whose components expansion in terms of the bi-vector
basis elements is f (a) =
∑D(D−1)/2
1
f
(a)
[n1n2]
e[n1n2].
The double-index notation in (6) just represents the bi-vector coordinates
character of the areas. Because there are D(D− 1)/2 independent bi-vectors in
D-dimensions, this explains the presence of the factor D(D−1)/2 in the denomi-
nator of eq-(6) and which coincides with the dim V of the real-valued irreducible
representation of the face-transitive group G′. The generalized Kronecker deltas
are
δ[n1n2] [n′1n′2] = δn1n′1δn2n′2 − δn1n′2δn2n′1
The extension of the Schur orthogonality relations for polyvectors is rigorously
analyzed in the Appendix. In D,D′-dimensions there are D(D− 1)/2, D′(D′ −
1)/2 bi-vector coordinates, respectively. Summing (tracing) over the double
indices [n1n2], restricting the summation over the bi-vector indices from 1 to
D′(D′ − 1)/2, and summing over the F group elements associated with one
representative of each one of the respective F cosets gH ′ corresponding to the
(normal) subgroup H ′ of G′ that leaves fixed a chosen face in eq- (6), gives
D′(D′−1)/2
∑
[n1n2]=1
F∑
1
R[n1n2] [12] R[n1n2] [12] =
D′(D′ − 1)
2
F
D(D − 1)/2
(7)
3
after following similar arguments as above, eq-(7) leads to the sums of areas
squared law ( A = 1⇒
∑
faces(A)
2 = F )
∑
faces (A
′)2
∑
faces(A)
2
=
D′(D′ − 1)
D(D − 1)
(8)
For example, lets take regular polyhedra inD = 3 and project down ontoD′ = 2.
Taking a cube with 6 sides of unit area, upon an orthogonal projection onto the
base comprised of one of its square faces , one has two squares of unit area (
the top and bottom), and four faces of zero area (the four side faces, the four
walls, yield a zero projection). Then the ratio of the sums of the areas squared
will be 26 =
1
3 =
2(2−1)
3(3−1) , which agrees with eq-(8).
In the case of a regular tetrahedron, one has upon an orthogonal projection
onto the equilateral triangular base, the following areas : the base triangle of
area equal to unity. Each one of the remaining 3 projections of the other 3
equilateral triangles will have an area equal to 13 of the area of the base triangle.
Their projections are 3 congruent triangles which fit inside the base triangle
without any gaps. So the ratio of the sums of the areas squared is
1 + 3( 13 )
2
4
=
1 + 13
4
=
4
3
4
=
1
3
=
2(2− 1)
3(3− 1)
=
D′(D′ − 1)
D(D − 1)
(9)
In the case of an octahedron involving 8 equilateral triangular faces of unit
areas, one has that the edge length L must be such that Area = L2
√
3
4 = 1 ⇒
L2 = 4√
3
. The projection onto the equatorial square section gives 2 copies of 4
triangles which fit inside the square of area L2 = 4√
3
. So each triangle will have
an equal area of 14L
2 = 1√
3
. The sum of the unit areas squared of the 8 faces of
the octahedron is 8. Thus the sums of the projected areas squared is then
2 × 4 [ 1
4
L2 ]2 = 8
1
3
⇒
∑
faces (A
′)2
∑
faces(A)
2
=
8
3
8
=
1
3
=
D′(D′ − 1)
D(D − 1)
(10)
In the case of a dodecahedron, with 12 pentagonal faces of unit area, one
has after the orthogonal projection onto a pentagonal base of unit area, the
following : The top and bottom base give an area squared equal to 1 + 1 = 2.
The remaining 10 faces are oriented such that the angles formed by the normals
to their faces with the vertical axis are given by α = arcos 1√
5
= 63.4349 =
180−116.5651 degrees. The angle 116.5651 degrees is the dihedral angle among
the pentagonal faces [4]. Therefore, the squared of the 10 projected areas yield
a net value of 10 cos2α = 105 = 2. Hence the sum of the projection of the 12
areas squared is then 2 + 2 = 4, and such that the ratio of the areas is 412 =
1
3
as expected.
Finally in the icosahedron case, one has 20 equilateral triangular faces than
can be grouped into 5 top faces, 10 middle faces and 5 lower faces. If the areas
4
of the 20 equilateral triangles are normalized to unity their edges must have for
length L such that Area = L2
√
3
4 = 1⇒ L
2 = 4√
3
. The projection of the 5 top
and 5 bottom faces onto two pentagonal horizontal cross sections yield a total
of 10 congruent triangular regions which fit inside the two pentagons. The areas
of the pentagonal horizontal cross section are
Area(pentagon) = 5
L2
4
1
tan(π/5)
= 5
4√
3
4
1
tan(π/5)
=
5
√
3 tan(π/5)
(11)
so the projection of the 5 bottom and 5 top triangles will each have an area
equal to 1/5-th the area of the above pentagon :
1
√
3 tan(π/5)
. So the sums of
the projections of these areas-squared will be
10 (
1
√
3 tan(π/5)
)2 = 10
1
3 tan2(π/5)
= 6.314
(12)
We are left with the projections of the 10 triangles in the middle region.
Their normals make an angle with respect to the vertical axis (orthogonal to
the pentagonal horizontal cross sections) given by β = 79.19 degrees. The
supplement angle is 180 − 79.19 = 100.81 degrees. The angle formed by the
normals to the 5 top and 5 bottom triangles with the vertical axis is
α = arcos(
1
√
3 tan(π/5)
) = 37.37 degrees
(13)
Therefore the net angle is 100.81 + 37.37 = 138.18 degrees which coincides with
the value of the dihedral angle of the icosahedron arccos(−
√
5/3)[4].
Therefore the sums of the squares of the projected areas of the 10 triangles
of the middle region of the icosahedron are
10 cos2(β) = 10 cos2(79.19) = 10 × 0.0352 = 0.352
(14)
Adding then eqs(12, 14) leads to a net value of 0.352 + 6.314 = 6.666 ∼ 203 .
Therefore the ratio of the areas is
20
3×20 =
1
3 as expected. An exact result would
be obtained if one writes all the expressions in terms of the Golden Mean (in
terms of quantities involving the
√
5). This completes our arguments for the 5
Platonic solids.
In the case of 3-dim cells and for a cell-transitive group G′′ one has for the
Schur orthogonality relation the following generalization of eq-(6) to the case of
generalized matrices R
(g)
[n1n2n3] [m1m2m3]
with trivector-valued indices
|G′′|∑
g
R
(g)
[n1n2n3] [123]
R
(g)
[n′1n
′
2n
′
3] [123]
= δ[n1n2n3] [n′1n′2n′3] δ[123] [123]
|G′′|
D(D − 1)(D − 2)/3!
(15)
5
The triple-index notation in (15) just represents the tri-vector coordinates char-
acter of the volumes. Because there are D(D − 1)(D − 2)/3! independent tri-
vectors in D-dimensions, this explains the presence of the factor D(D− 1)(D−
2)/3! in the denominator of eq-(15) and which coincides with the dim V of the
real-valued irreducible representation of the 3-dim cell-transitive group G′′. The
generalized Kronecker deltas in (15) are defined by the determinant
δ[n1n2n3] [n′1n′2n′3] = det
 δn1n′1
. . .
δn1n′3
δn2n′1
. . .
δn2n′3
δn3n′1
. . .
δn3n′3

(16)
Repeating the same argument as before, if instead of edge/face transitivity one
assumes now cell-transitivity, one arrives at the ratio
∑
cells (V
′)2
∑
cells(V )
2
=
D′(D′ − 1)(D′ − 2)
D(D − 1)(D − 2)
(17)
and in general for higher dimensional cells, p-dimensional hyper-volumes , one
has the ratio∑
hypercells (V ′)2
∑
hypercells(V)2
=
D′(D′ − 1)(D′ − 2) ..... (D′ − p+ 1)
D(D − 1)(D − 2) ..... (D − p+ 1)
(18)
Acknowledgements
We are indebted to Garret Sadler for extensive numerical computations cor-
roborating the sums of lengths (areas) squares associated to a very large number
of polyhedra, and to J. Kovacs, T. Smith and M. Pitkanen for discussions. One
of us, C. C. P wishes to thank M. Bowers for very kind assistance and to the
Quantum Gravity Research Group for their generous support.
Appendix : Schur Orthogonality Relations for Polyvectors
In D = 3 there is a one-to-one correspondence between vectors A = Aiei =
Aiγi and bivectors B = Bjke
jk = Bjke
j∧ek = Bjkγj∧γk. In particular they are
duals of each other : Ai = ijkBjk. Therefore, one can rotate (transform) the
bivectors using the generalized matrices with multi-indices such that B′[j1j2] =
R[j1j2] [k1k2]B[k1k2] (recurring to Einstein’s summation convention for repeated
indices), or one can rotate (transform) their dual vectors (the normals to the
planes associated to the bivectors) such that A′j = RjkAk. The one-to-one
correspondence between the matrices and generalized matrices’ components is
R[12] [12] ↔ R33, R[13] [13] ↔ R22, R[23] [23] ↔ R11
R[12] [13] ↔ −R32, R[12] [23] ↔ R31, R[23] [12] ↔ R13, ........
(A.1)
The face-transitive group G′ in this case will map the normal vector of one
face into the normal vector of another face; which is equivalent to mapping
6
the bivector of one face into the corresponding bivector of another face of the
polyhedron. Therefore, in D = 3 one can implement the Schur orthogonality re-
lations involving unitary irreducible representations of the face-transitive group
G′ , either by using the ordinary 3×3 matrices Rjk, or by using their associated
generalized matrices R[j1j2] [k1k2] with bivector indices in (A.1), as follows
|G′|∑
g
R(g)
nm R
∗(g)
n′m′ = δnn′ δmm′
|G′|
dim V
←→
|G′|∑
g
R
(g)
[n1n2] [m1m2]
R
∗(g)
[n′1n
′
2] [m
′
1m
′
2]
= δ[n1n2] [n′1n′2] δ[m1m2] [m′1m′2]
|G′|
dim V
(A.2)
where in our case described above one has in D = 3 that the dim V = 3.
In higher dimensions, because there are more than one normal vector to a
given surface, the correct way to proceed is as follows. If the symmetry group
is edge-transitive (face-transitive) it implies that it maps an edge (face) into
another edge (face). Given two adjacent (incident) and directed edges ~e(1), ~e(2)
at a vertex v of a face of a regular polytope in D > 3, the bivector ~e(1) ∧ ~e(2)
associated with its face has for magnitude a value proportional to its area. In
the case of the cube in D = 3, the proportionality constant for a square face
is unity. In the case of a pentagonal face (dodecahedron) the proportionality
constant is 4cos(3π/5)
5
.
If there is a symmetry group G′ which is face-transitive it means that one
can map a bivector ~e(1) ∧~e(2) to another bivector ~e(α) ∧~e(α+1) lying on another
face. The indices α, α+1 denote two other adjacent (incident) edges at a vertex
in another face of the regular polytope. Furthermore, if the point group G of
the polytope is edge-transitive one has that
~e(α) = R(α,1)
n1m1 ~e
(1)
m1 e
n1 , ~e(α+1) = R(α+1,2)
n2m2
~e(2)m2 e
n2
(A.3)
where once again we employ the Einstein summation convention : a summation
of indices is performed over the repeated indices. Given that
~e(2) = R(2,1)
n3m3 ~e
(1)
m3 e
n3
(A.4)
one can rewrite
~e(α+1) = R(α+1,2)
n2m2
~e(2)m2 e
n2 = R(α+1,2)
n2m2 R
(2,1)
m2m3 ~e
(1)
m3 e
n2 =
R(α+1,1)
n2m3
~e(1)m3 e
n2 , since R(α+1,1)
n2m3 = R
(α+1,2)
n2m2 R
(2,1)
m2m3
(A.5)
resulting from the group composition law and definition of a group represen-
tation R(g)R(g′) = R(g · g′). From eqs-(A.3-A.5) we infer that the bivector
~e(α) ∧ ~e(α+1) can be expressed as
~e(α) ∧ ~e(α+1) = [ R(α,1)
n1m1 ~e
(1)
m1 e
n1 ] ∧ [ R(α+1,1)
n2m3
~e(1)m3 e
n2 ]
(A.6)
7
Choosing the first coordinate axis to coincide with the direction of ~e(1) implies
that its only nonvanishing component is ~e
(1)
1 = 1, so the bivector in (A.6)
becomes
~e(α) ∧ ~e(α+1) = [R(α,1)
n11
en1 ] ∧ [R(α+1,1)
n21
en2 ] = [R
(α,1)
n11
R
(α+1,1)
n21
] en1 ∧ en2 =
1
2
[ R
(α,1)
n11
R
(α+1,1)
n21
− R(α,1)
n21
R
(α+1,1)
n11
] e[n1n2], e[n1n2] ≡ en1 ∧ en2 (A.7)
In the last line of (A.7) we have antisymmetrized the expression with respect to
the indices n1, n2 due to the antisymmetry property of the bivector indices. The
contribution of the symmetric terms is zero. Hence, from (A.7) we can finally
read-off the components of the generalized matrix R(a1), comprised of bivector
indices, in terms of the ordinary matrix components with vector indices and
representing the map which sends the face f (1) of unit area into another face
f (a) of unit area
R
(a1)
[n1n2] [12]
=
κ
2
[ R
(α,1)
n11
R
(α+1,1)
n21
− R(α,1)
n21
R
(α+1,1)
n11
]⇒
f (a) = R
(a1)
[n1n2] [12]
f
(1)
[12] e
[n1n2] = R
(a1)
[n1n2] [12]
e[n1n2]
(A.8)
since f
(1)
[12] = 1, after orienting the face f
(1) along the bivector e[12], and where
κ is a numerical constant whose value depends on the shape of the faces; i.e.
squares, triangles, pentagons. For example, κ = 1 for squares, κ = 4cos(3π/5)
5
for pentagons, ...
In (A.8) there is a correlation between the edge index α
enumeration and the face index a enumeration of the polytope. Correlation
which can always be found in any polytope. In the case of the cube, there are
3 incident edges at a common vertex within each one of the 6 faces (planes).
The independent bivectors are e12, e13, e23 and correspond to the 3 plane classes
(orientations) of the cube in 3-dimensions.
Concluding, the Schur orthogonality relations of the face-transitive group G′,
for unitary irreducible representations of dimension D(D− 1)/2, given in terms
of generalized matrices R(a1) and whose components are expressed in terms of
bivector indices R
(a1)
[n1n2] [12]
as displayed explicitly in eq-(A.8), are given by
|G′|∑
g
R
(g)
[n1n2] [m1m2]
R
∗(g)
[n′1n
′
2] [m
′
1m
′
2]
= δ[n1n2] [n′1n′2] δ[m1m2] [m′1m′2]
|G′|
dim V
(A.9)
with dim V = D(D − 1)/2. In the case of real-valued representations one has
R
∗(g)
[n′1n
′
2] [m
′
1m
′
2]
= R
(g)
[n′1n
′
2] [m
′
1m
′
2]
. One can repeat the above arguments for the
higher grade polyvectors corresponding to volumes, hypervolumes, .... leading
to eqs-(17,18) for cell-transitive and hyper-cell-transitive groups.
8
References
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W. Miller, Jr., Symmetry Groups and their Applications, (Academic
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[4] R. Williams, The Geometrical Foundations of Natural Structure (Dover
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9