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arXiv:2303.03452v1 [math.GM] 6 Mar 2023
Geometric Algebras of Light Cone Projective
Graph Geometries
Garret Sobczyk
Departamento de Actuaría Física y Matemáticas,
Universidad de las Américas-Puebla,
72820 Puebla, Pue., México
garretudla@gmail.com
March 8, 2023
Abstract
A null vector vis an algebraic quantity with the property that v2= 0.
I denote the universal algebra generated by taking all sums and products
of null vectors over the real or complex numbers by N, [1]. The rules
of addition and multiplication in Nare taken to be the familiar rules of
addition and multiplication of real or complex square matrices. A pair of
null vectors is positively or negatively correlated if their inner product is
positive or negative, respectively. A large class of geometric algebras are
isomorphic to real or complex 2n×2nmatrix algebras, or a pair of such
algebra. I begin the study of the eigenvector-eigenvalue problem of linear
operators in the geometric algebra G1,n of Rn+1, and by restricting to
barycentric coordinates, n-simplices whose n+1 vertices are non-zero null
vectors. These ideas provide a foundation for a new Cayley-Grassmann
Theory of Linear Algebra, with many possible applications in pure-applied
areas of science and engineering.
AMS Subject Classification: 03B30,05C20,15A66,15A75
Keywords: Clifford algebra, complete graphs, Grassmann algebra, Lorentzian
spacetime.
0. Introduction
The origin of the ideas in this paper date back most directly to mathematics
that was set down in the nineteenth century by H. Grassmann [2], A. Cayley
(Memoir on the Theory of Matrices 1858), and W. Clifford [3]. It is regrettable
today, after more than 150 years, that Clifford’s geometric algebra has not found
it proper place in the Halls of Mathematics and Science [4]. My journey in this
saga began in 1965, when I starting working in geometric algebra as a graduate
Ph.D. student of Professor David Hestenes at Arizona State University [5], and
continued with years spent with gracious colleagues in Poland and Mexico. It
1
is my belief that this paper will bring us closer to the day when geometric
algebra has finally found its proper place in the Millennial Human Quest for the
development of the geometric concept of number [6].
In Section 1, it is shown that the geometric algebras G1,n and Gn,1have spe-
cial bases of all positively, or all negatively correlated null vectors, respectively.
In the case of G1,n , the inner products can all be chosen to be +1
2, and in the
case of Gn,1,−1
2. For simplicity, the classification of endomorphisms on Rn+1
is considered only in the case of a (+ 1
2)-positively correlated basis of a geomet-
ric algebra G1,n, but the same analysis is valid for studying endomorphisms on
Rn+1 of a (−1
2)-negatively correlated basis of a geometric algebra Gn,1.
In Section 2, basic ideas of linear algebra in Rn+1 are developed in the
symmetric algebra A+
n+1 of a correlated basis of null vectors in G1
1,n. The
concept of a LPGG star projection of a geometric number is defined and studied.
The vector derivative ∇is defined, paying particular attention to its important
properties.
In Section 3, basic properties of lower dimensional geometric algebras are ex-
plored in the correlated basis algebra A+
3of G1,3. The concept of the LPGG star
projection suggests that a new classification all geometric algebras is possible
in the correlated null vector algebra A+
n+1 of G1,n, [7].
In Section 4, by introducing barycentric coordinates, complete graphs are
studied in which every pair of vertices is connected by an edge. Light Cone
Projective Geometry (LPGG) is built upon the property that for any dimension
n≥1, there exits positively, or negatively correlated light cones, defined by
sets of (n+ 1) null basis vectors {a1,...,an+1}of G1,n , or Gn,1, such that
a1∧ · ·· ∧ an+1 6= 0 and ai·aj=±(1−δij )
2, respectively.
1 The geometric algebras G1,n and Gn,1of Rn+1
The geometric algebras G1,n and Gn,1arise from null vector bases of Rn+1 by
constructing positively, or negatively correlated, null vectors in terms of the
standard bases {e1, f1,···, fn}of G1,n, or {f1, e1,···, en}of G1,n, respectively.
Renewed interest in these Clifford algebras is due in part to the pivotal Lecture
Notes published by Marcel Riesz in 1958, [8]. The geometric algebras G1,n and
Gn,1make up the two fundamental sequences of successively larger algebras,
R⊂G1,1⊂G1,2⊂G1,3⊂ ··· ⊂ G1,n ⊂ ··· ⊂ N,(1)
and
R⊂G1,1⊂G2,1⊂G3,1⊂ ··· ⊂ Gn,1⊂ ··· ⊂ N,(2)
where Nis the universal algebra generated by taking sums and products of null
vectors. See [1, 6, 9], and other references.
Let {a1,...an+1 } ⊂ Rn+1 be a set of positively, or negatively, correlated null
vectors satisfying a1∧ · ·· ∧ an+1 6= 0, and the (n+ 1)2properties,
ai·aj≡1
2(aiaj+ajai) := ±1−δij
2for 1 ≤i, j ≤n+ 1,(3)
2
respectively, where δij is the usual delta function. In terms of these basis null
vectors,
Rn+1 := {x|x=x1a1+···+xn+1 an+1, xi∈R}.(4)
The multiplication tables for sets of positively (PC), or negatively (NC), corre-
lated null vectors ai, aj, for 1≤i < j ≤n+1, follow directly from the properties
(3), and generate the positively, and negatively correlated null vector algebras
A+
1,n =G1,n, and A−
n,1=Gn,1, respectively.
Table 1: Multiplication Table.
aiajaiajajai
ai0aiaj0ai
ajajai0aj0
aiajai0aiaj0
ajai0aj0ajai
For a set of positively or negatively correlated null vectors {a1,...,an},
define
Ak:=
k
X
i=1
ai.(5)
The geometric algebra
G1,n := R(e1, f1,...,fn),
where {e1, f1, . . . , fn}is the standard basis of anticommuting orthonormal vec-
tors, with e2
1= 1 and f2
1=··· =f2
n=−1. The 2n+1-canonical forms of the
standard multivector basis elements are
n1; e1, f1,···, fn;e1f1,···, e1fn,1≤i<k≤nfifk,;...;e1f1···fno.(6)
Alternatively, the geometric algebra G1,n can be defined by
G1,n := R(a1,...,an+1 ) =: A+
1,n,
where {a1,...,an+1 }is a set of positively correlated null vectors satisfying the
Multiplication Table 1. In this case, the standard basis vectors of G1,n can be
defined by e1=a1+a2=A2,f1=a1−a2=A1−a2, and for 2≤k≤n
fk=αkAk−(k−1)ak+1,(7)
where αk:= −√2
√k(k−1) . The 2n+1 -canonical forms of the standard multivector
basis elements of A+
n+1 are
n1; a1,...,an+1 ;1≤i<j≤n+1 aiaj,;...;a1···an+1o.(8)
3
The geometric algebra
Gn,1:= R(f1, e1,...,en),
where {f1, e1,...en}is the standard basis of anticommuting orthonormal vec-
tors, with f2
1=−1and e2
1=···=e2
n= 1. Alternatively, the geometric algebra
Gn,1can be defined by
Gn,1:= R(a1,...,an+1 ) =: A−
1,n,
where {a1,...,an+1 }is a set of negatively correlated null vectors satisfying the
Multiplication Table 2. In this case, the standard basis vectors of Gn,1can be
defined by f1=a1+a2=A2,e1=a1−a2=A1−a2, and for 2≤k≤n
ek=αkAk−(k−1)ak+1,(9)
where αk:= −√2
√k(k−1) . The 2n+1 -canonical forms of the standard multivector
basis elements of A−
n+1 is the same as (8).
Table 2: Multiplication Table.
aiajaiajajai
ai0aiaj0−ai
ajajai0−aj0
aiaj−ai0−aiaj0
ajai0−aj0−ajai
For the remainder of this paper, only properties of the positively correlated
null vector algebras A+
n+1 := A1,n of the geometric algebras of G1,n are con-
sidered. It should be recognized, however, that any of these properties can be
easily translated to the corresponding properties of the negatively correlated
null vector basis algebras A−
n+1 := An,1of Gn,1. Indeed, much more general
algebras of correlated null vectors in Ncan be defined and studied, but with
correspondingly more complicated rules of multiplication. In addition to provid-
ing a new framework for the study of Linear Algebra on Rn+1, the last section
of the paper shows how the ideas can be applied to graph theory.
2 Linear algebra of Rn+1 in A+
n+1
The position vector x∈Rn+1 in the standard basis of G1,n is
x:= s1e1+
n
X
i=1
si+1fi∈G1
1,n.(10)
4
Alternatively, in the correlated null vector basis algebra A+
n+1 =G1,n,
x=
n+1
X
i=1
xiai∈ A+
n+1.(11)
Since geometric algebras are fully compatible with matrix algebras, matrix
algebras over geometric algebras are well defined [6]. To relate the bases (10)
and (11), in matrix notation
x= ( s1. . . sn+1 )
e1
f1
·
·
·
fn
= ( x1. . . xn+1 )
a1
a2
·
·
·
an+1
,(12)
or in abbreviated form,x=s(n+1)F(n+1) =x(n+1)A(n+1) . The quadratic form
of G1,n is specified by F−1
(n+1) := Ft
(n+1)B, where
B:= F(n+1) ·Ft
(n+1) =
e1
f1
·
·
·
fn
·(e1f1. . . fn)(13)
=
1 0 0 ··· 0
0−1 0 ··· 0
· · · · ·
· · · · ·
0 0 ··· 0−1
,
where Ft
(n+1) denotes the row transpose of the column F(n+1).
Let v, w ∈G1
1,n be vectors. Expressed in the standard basis of G1,n, the
geometric product
vw =v(n+1) F(n+1)Ft
(n+1)wt
(n+1)
=v(n+1)F(n+1) ·Ft
(n+1)wt
(n+1) +v(n+1)F(n+1) ∧Ft
(n+1)wt
(n+1)
=v(n+1)Bwt
(n+1) +v(n+1)
0e1f1e1f2··· e1fn
f1e10f1f2··· f1fn
· · · · ·
· · · · ·
fne1fnf1fnf2··· 0
wt
(n+1).(14)
Dotting each side of the equation (12) on the right by the row matrix
F−1
(n+1) := ( e1−f1... −fn),
5
and noting that
e1
f1
·
·
·
fn
·(e1−f1... −fn)
is an expression for the (n+ 1) ×(n+ 1) identity matrix, immediately gives
s(n+1) =x(n+1)T, where the matrix of transition Tis defined by the Gramian
matrix
T:= A(n+1) ·F−1
(n+1) =
a1
a2
·
·
·
an+1
·(e1−f1... −fn)(15)
in terms of the inner products ai·fj. These inner products are directly calculated
using (7). The transition matrix T8, and its inverse T−1
8, for the geometric
algebra G1,7is given in Appendix A.
Note, that whereas Tis the transition matrix
T F(n+1) =T
e1
f1
·
·
fn
=
a1
·
·
an+1
=A(n+1),(16)
of the column basis vectors F(n+1) to the column basis vectors A(n+1) ,T−1is
the coordinate transition matrix
s(n+1)T−1= ( s1··· sn+1 )T−1= ( x1··· xn+1 ) = x(n+1),(17)
from the row vector coordinates of xto the row vector coordinates of xin the
basis A(n+1). Great care must be taken to avoid confusion.
Converting the calculation in (14) to a calculation for v, w ∈ A+
n+1,
vw =v(n+1) F(n+1)Ft
(n+1)wt
(n+1)
=v(n+1)T−1T F(n+1) Ft
(n+1)Tt(Tt)−1wt
(n+1)
=va
(n+1)A(n+1) At
(n+1)(wa
(n+1))t,
giving
vw =va
(n+1)A(n+1) ·At
(n+1)(wa
(n+1))t+va
(n+1)A(n+1) ∧At
(n+1)(wa
(n+1))t
=va
(n+1)
0a1a2a1a3··· a1an+1
a2a10a2a3··· a2an+1
· · · · ·
· · · · ·
an+1a1an+1 a2an+1a3··· 0
(wa
(n+1))t.(18)
6
2.1 Bivector endomorphisms in A+
n+1
The standard treatment of the relationship between Clifford’s geometric algebras
and Cayley’s matrix algebras is well-known, [9, p.74], [10, p.217]. Something
that has always disturbed me is that this relationship is an isomorphism only
for square matrix algebras of order 2n×2n. This sorry state of affairs is at least
partially rectified in the algebras A+
n+1 =G1,n.
Recalling the definition of (5), it is not difficult to show that for k≥2,
(Ak)2=A2
k−1+ 2k
X
j=2
A2
j−1·aj= (k−1) + ···+ 1 = k
2.(19)
Defining ˆ
Ak:= √2
√k(k−1) Ak, it follows that ˆ
A2
k= 1. For g∈ A+
n+1, and k > 1, I
now define the LPGG k-projection of g,
g⋆:= ˆ
Akgˆ
Ak.(20)
When k=n+ 1, the ⋆-projection becomes a conjugation on G1,n.
Clearly, for all g∈ A+
n+1,(g⋆)⋆=g, and given a second h∈ A+
n+1,
(gh)⋆=ˆ
An+1gˆ
An+1 ˆ
An+1hˆ
An+1 =g⋆h⋆.(21)
Defining the A-matrix of g∈ A+
n+1,
[g]a:= A(n+1)gAt
(n+1) = [aigaj]a,(22)
it follows that
g⋆=ˆ
An+1gˆ
An+1 =It
(n+1)[g]aI(n+1) ,
where I(n+1) and It
(n+1) are the n+ 1 column and row matrices
I(n+1) :=
1
·
·
·
1
,It
(n+1) := ( 1 1 ··· 1 ) ,
respectively.
The GA product of g, h ∈ A+
n+1, in terms of their (n+1)×(n+ 1) A-matrices
[g]a,[h]a, then takes the unusual form
gh =ˆ
An+1It
(n+1)[g]aI(n+1) It
(n+1)[h]aI(n+1) ˆ
An+1,(23)
mediated by the square singular (n+ 1)-matrix I(n+1)It
(n+1). Equation (23) is
a generalization of the closely related formula (18) for the multiplication of the
vectors v, w ∈ A+
n+1.
7
It follows from (20) and (23) that a real or complex (n+ 1) ×(n+ 1)-matrix
[gij ]is the matrix of a scalar plus a bivector g∈ A+
n+1, that is
[g]a=ˆ
An+1[g⋆]aˆ
An+1
=
0g12a1a2g13 a1a3··· g1,n+1a1an+1
g21a2a10g23 a2a3··· g2,n+1a2an+1
· · · · ·
· · · · ·
g1,n+1an+1 a1g2,n+1an+1 a2gn,n+1an+1 a3··· 0
.
(24)
Comparing the matrix [g]ain (24) to the matrix in (14), seems to contradict
that the trace of a matrix is invariant under a change of basis. However this
is not the case since the terms gijaiajof (24) consists of scalars and bivectors.
Plücker relations are important in understanding the structure of bivectors [11],
particularly bivectors in G1,n, and in study of conformal mappings [12].
2.2 The gradient ∇
A crucial tool for carrying out calculations in the geometric algebra G1,n is the
gradient ∇. In the references [5, 6, 13], the gradient ∇, alongside the geometric
algebra Gn, has been developed as a basic tool for formulating and proving
basic theorems of linear algebra in Rn. Since the properties of the gradient
are independent of the quadratic form of the geometric algebra used, instead of
using the Euclidean geometric algebra Gn+1 of Rn+1, we can equally well define
it in terms of the geometric algebra G1,n . It follows that all theorems of linear
algebra developed in [5, 6, 13] are equally valid in G1,n without modification.
In the standard basis of G1,n ,
∇:= e1
∂
∂s1−f1
∂
∂s2− · · · − fn
∂
∂sn+1
.(25)
With the transition matrix (15) in hand, the expression for the gradient in
the null vector basis of A1,n,
∇=
n+1
X
i=1
(∇xi)∂
∂xi
,(26)
is nothing more than a simple expression of the chain rule in calculus. In terms of
the abbreviated notation for (12), it is not difficult to derive the transformation
rules relating the bases columns A(n+1) and F(n+1).
Using (15), and solving
x=s(n+1)F(n+1) =x(n+1) A(n+1),(27)
gives the important relations
•s(n+1) =x(n+1)T⇐⇒ x(n+1) =s(n+1)T−1
8
•x·F−1
(n+1) =s(n+1) =x(n+1)T⇐⇒ F−1
(n+1) =∇s(n+1) =∇x(n+1)T
•x(n+1) =s(n+1)T−1⇐⇒ A−1
(n+1) := ∇x(n+1) =F−1
(n+1)T−1.
•A−1
(n+1) =F−1
(n+1)T−1⇐⇒ A−1
(n+1)T=F−1
(n+1).
•∇x=n+ 1 = F−1
(n+1)F(n+1) =A−1
(n+1)A(n+1) .
•F(n+1) =F(n+1) · ∇x=F(n+1) · ∇x(n+1)A(n+1)
=F(n+1) ·∇s(n+1) T−1A(n+1) =T−1A(n+1).
Note, whereas F(n+1) and A(n+1) have been defined as column matrices of
vectors, F−1
(n+1) and A−1
(n+1) are row matrices of vectors. Taking the outer product
of basis vectors in the relation F(n+1) =T−1A(n+1), gives
∧F(n+1) = det T−1∧A(n+1),
or equivalently, after calculating and simplifying,
e1f1···fn=−(√2)n+1
√na1∧ · ·· ∧ an+1,(28)
relating the pseudoscalar elements of the geometric algebra G1,n expressed in
the standard basis and in the null vector basis of A+
n+1.
2.3 Decomposition formulas for ∇
Usually the concept of duality is defined in terms of the operation of multiplica-
tion in an algebraic structure. By defining the geometric algebras G1,n and Gn,1
in terms of the null vector basis algebras A+
1,n and A−
n,1, whose rules of multipli-
cation have been given in the Multiplication Tables 1 and 2, suggests defining
the concept of duality in terms of addition. The dual n-sum ∨
aiof ai∈ A+
n+1 is
the n-sum
∨
ai:= a1+···+∨
i+···an+1,(29)
formed leaving out the ith term of the basis null vectors {a1,...,an+1} ⊂ A+
n+1.
Calculations with the gradient ∇in A+
n+1 can often be simplified using the
following decompostion formulas. Defining the dual sum and null gradients
∨
∇:=
n+1
X
i=1
∨
ai∂iand ˆ
∇:=
n+1
X
i=1
ai∂i,(30)
respectively, the gradient ∇defined by (26), ∨
∇and ˆ
∇defined in (30), satisfy
the following decomposition formulas:
•∇=2
nAn+1∂(n+1) −nˆ
∇=2
n∨
∇ − (n−1) ˆ
∇,
where ∂(n+1) := Pn+1
i=1 ∂i.
9
•An+1 · ∇ = (n+ 1)∂(n+1) −2An+1 ·ˆ
∇
•∨
∇+ˆ
∇=An+1∂(n+1) ⇐⇒ An+1 ·∨
∇+An+1 ·ˆ
∇=(n+1)n
2∂(n+1)
•ˆ
∇2=Pn+1
i<j ∂i∂j,∨
∇2=(n+1)n
2Pn+1
i=1 ∂2
i+ (n2−n+ 1) Pn+1
i<j ∂i∂j
•∇2=4
n2∨
∇ − (n−1) ˆ
∇2=∨
∇2−2(n−1)∨
∇· ˆ
∇+ˆ
∇2,
where ∨
∇ · ˆ
∇=n
2
∨
∇∂(n+1) −ˆ
∇2.
Verifications of the above formulas, which are omitted, depend heavily on the
combinatorial-like identities
A2
n+1 =(n+ 1)n
2, ai·An+1 =An+1 ·ai=n
2∨
ai,(31)
and the additive duality formula for An+1, and 1≤i < j ≤n+ 1,
∨
ai·∨
aj=n2−n+ 1.(32)
3 Lower dimensional geometric algebras
This section characterizes geometric sub-algebras of A+
3≡G1,2in R3.
The pseudoscalar
i:= e1f1f2=−2a1∧a2∧a3,(33)
is in the center of the algebra, commuting with all elements. The algebra
G3:= R(e1, e2, e3),
is obtained from the algebra G1,2, simply by defining e2=if1=e1f2∈G2
1,2
and e3=−if2=e1f1∈G2
1,2, and reinterpreting these anticommuting elements
to be vectors in G1
3.
The matrix coordinates [e1],[e2],[e3]of e1, e2, e3, known as the famous Pauli
matrices, opened the door to the study of quantum mechanics [9, p.108]. It
has found many recent applications in computer science and robotics, [17]. The
geometric algebra G3is isomorphic to the even subalgebra of the spacetime
algebra G1,3=A+
4of R4. Its matrix version is known as the Dirac algebra.
The null vector basis algebra A+
3=G1,2is defined by 3null vectors {a1, a2, a3}
with the property that ∧A(3) 6= 0,ai·aj=(1−δij )
2, and the Multiplication Table
1. The relations between the standard basis of G1,2, and the basis of A+
3, are
summarized by the 3×3transition matrix T3, and its inverse,
T3:=
1
2
1
20
1
2−1
20
1 0 1
, T −1
3:=
1 1 0
1−1 0
−1−1 1
.(34)
10
Using the relations given after (27),
e1
f1
f2
=T−1
3
a1
a2
a3
=
a1+a2
a1−a2
−a1−a2+a3
(35)
and
a1
a2
a3
=T3
e1
f1
f2
=
1
2(e1+f1)
1
2(e1−f1)
e1+f2
.(36)
The canonical forms relating the vectors, bivectors and trivectors are:
•e1=a1+a2, f1=a1−f1, f2=−a1−a2+a3
•e1f1= (a1+a2)(a1−a2) = a2a1−a1a2= 1 −2a1a2
•e1f2= (a1+a2)(−a1−a2+a3) = 1 + a1a3−a2a3
•e1f1f2= (a1+a2)(1 −a1a2+a1a3−a2a3) = a1+a3−2a1a2a3
One of the simplest endomorphisms, f:R2→R2, defined by v1, v2∈R2, is
f(x) := 2(v1∧v2)x= 2(x·v2)v1−(x·v1)v2,(37)
where vi=vi1a1+vi2for i∈ {1,2}. The endomorphism f(x)has the eigenvec-
tors a1and a2, with the eigenvalues ±det v11 v12
v12 v22 ,
f(a1) = 2(v1∧v2)a1= det v11 v12
v12 v22 a1,(38)
f(a2) = 2(v1∧v2)a2=−det v11 v12
v12 v22 a2,(39)
respectively, as is easily verified.
Now calculate,
(v1∧v2)(v1∧v2∧x) = (v1∧v2)·(v1∧v2∧x)
= (v1∧v2)2x+ (v1∧v2)·(v2∧x)v1+ (v1∧v2)·(x∧v1)v2= 0.
Dividing both sides of this last equation by (v1∧v2)2, gives
(v1∧v2∧x)
(v1∧v2)=x−(x∧v2)
(v1∧v2)v1+(x∧v1)
(v1∧v2)v2= 0,(40)
expressing the position vector x∈R2uniquely in terms of its LPGG projective
coordinates. Of course, the trivector v1∧v2∧x= 0, because we are in the geo-
metric algebra G1,1of R2. Multiplying equation (40) by 4(v1∧v2)2immediately
gives what I call the Cayley-Grassmann identity,
4(v1∧v2)(v1∧v2∧x) = f2(x)−4(v1∧v2)(x∧v2)v1+ 4(v1∧v2)(x∧v1)v2
11
=f2(x)−2f(x)·v2v1+ 2f(x)·v1v2= 0.(41)
The matrix of [f(x)] of f(x), in this translation, is given by
[f(x)] = [2(v1∧v2)x] = 2[(v1∧v2)][x],(42)
is the product of the matrix
[v1∧v2] = 1
2[v1v2−v2v1]=1
2[v1][v2]−[v2][v1],(43)
where [v1∧v2],[v1],[v2]are the matrices of v1∧v2, v1, v2, respectively, and [x]is
the matrix of x. These matrices are given below. With (35) and (36) in hand,
the matrix [x]of the position vector x∈R3
[x] = x1[a1] + x2[a2] + x3[a3] = x3i x2−x3
x1−x3−x3i
with respect to the basis A+
3, and
[x] = s1[e1] + s2[f1] + s3[f2] = s3i s1−s2
s1+s2−s3i
with respect to the standard basis of G1,2.
The 2×2matrices are defined with respect to the spectral basis
a2a1a2
a1a1a2
of G1,1, as detailed in [9] and [6, p.78]. The matrices of [a1]and [a2]of a1and
a2, are
[a1] = 0 0
1 0 ,[a2] = 0 1
0 0 ,
respectively, and
[x] = 0x2
x10,[v1] = 0v12
v11 0,[v2] = 0v22
v21 0,
which are used with (43) to calculate
[v1∧v2] = 1
2[v1][v2]−[v2][v1]=1
2v12v21 −v11 v22 0
0v11v22 −v12 v22 ,
and
[f(x)] = 2[(v1∧v2)][x] = 0 (v12v21 −v11 v22)x2
−(v12v21 −v11 v22)x10.
Unlike the usual representation of an endomorphism f(x)on R2, as a 2×2
matrix of [f]of f(x)times the column matrix of x,[x] = x1
x2, the matrix
12
of the endomorphism f(x)in the LPGG of V+
2(v1, v2)comes as the single real
matrix [f(x)]. This single matrix [f(x)] in LPGG can be broken into the product
of two 2×2matrices. By (42),
[f(x)] = 2[(v1∧v2)][x] = v12v21 −v11 v22 0
0−(v12v21 −v11 v22) 0x2
x10.
For k∈ {1,2,3}, let
vk:= vk1a1+vk2a2+vk3a3∈R3,
consider the endomorphism
f:R3→ A+
3,(44)
defined by
f(x) := 2(v1∧v2∧v3)x= 2 det[vij ](a1∧a2∧a3)x
= det[vij ](x1+x2)a1∧a2+ (x2+x3)a2∧a3+ (x1+x3)a3∧a1.
It is interesting to note that each of the bivectors in the above expression are
anticommutative and square to 1
4. In view of (33), this is not surprising. Indeed,
the mapping (44) can simply be expressed as the duality relation f(x) = −ix. It
follows that over the complex numbers, every vector x∈ A+
3is an eigenvector.
Consider the mapping g:R3→ A+
3, defined by
g(x) := ( 1 1 1 )
0g12a1a2g13 a1a3
g21a2a10g23 a2a3
g31a3a1g32 a3a20
1
1
1
x=Gx, (45)
where
G=1
2tr(G) + g1a2∧a3+ g2a3∧a1+ g3a1∧a2,(46)
for g1:= (g23 −g32),g2:= (g31 −g13),g3:= (g12 −g21), and
tr(G) := g12 +g13 +g21 +g23 +g31 +g32.
The same mapping (45) can equally well be considered over C,
g:C3→ A+
3(C),(47)
giving a new relationship between the Pauli matrices and G3. In this case,
e1:= 1
2a2∧a3, e2:= 1
2a3∧a1, e3:= 1
2a1∧a2.
The minimal polynomial of Gis easily calculated:
ϕ(G) = G−1
2tr(G)2−1
4g12+ g22+ g32.
13
Setting ϕ(r) = 0 and solving for r, gives the two roots r−and r+,
r∓:= 1
2tr(G)∓qg2
1+ g2
2+ g2
3.
In the spectral basis,Gtakes the form
G=r−p1(G) + r+p2(G),(48)
where
p1(t) := −2t+tr(G) + pg2
1+ g2
2+ g2
3
2pg2
1+ g2
2+ g2
3
and
p2(t) := 2t−tr(G) + pg2
1+ g2
2+ g2
3
2pg2
1+ g2
2+ g2
3
,
[6, 9, 14].
4 Simplices in A+
n+1
It has been shown in previous sections how the development of linear algebra
can be carried out in Rn+1, using the tools of G1,n ≡ A+
n+1. Restricting to
barycentric coordinates, gives new tools for application in graph theory. In
Simplicial Calculus with Geometric Algebra, many ideas of simplicial geometry
were set down in the context of geometric algebra [15]. The present work is in
many ways a continuation of this earlier work.
Let A+
n+1 be the null vector algebra of the geometric algebra G1,n, defined
by the Multiplication Table 1, and where the null vectors aisatisfy for 1≤i, j ≤
n+ 1,
ai·aj=(1 −δij )
2.(49)
For x∈Rn+1 the position vector (11), the convex null n-simplex in Rn+1 is
defined by
S+
n:= S+
n(a1,...,an+1 ) = {x∈Rn+1|x1+···+xn+1 = 1, xi≥0},(50)
by the requirement that the coordinates x(s)of x∈Rn+1 , are homogeneous
barycentric coordinates, [16].
By the content of S+
n, we mean
a△n:= 1
n!∧(n+1)
i=2 (ai−a1) = (a2−a1)∧(a3−a1)∧ · ·· ∧ (an+1 −a1)
=1
n!∧∨
a1− ∧∨
a2+···+ (−1)n∧∨
an+1.(51)
Wedging (51) on the left by x∈ S+
n, gives
x∧a△n=1
n!n+1
X
i=1
xi∧A(n+1) =1
n!∧A(n+1).(52)
14
Similarly, dotting (51) on the left by xgives
x·a△n=1
n!x·∧∨
a1− ∧∨
a2+···+ (−1)n∧∨
an+1.(53)
Let v1,...,vk+1 ∈Rk+1 be a set of k+ 1 vertices of a k-simplex V+
kin A+
k+1.
That is
vi:=
k+1
X
j=1
vij aj= [vij]A(k+1) ,(54)
where [V]k+1 := [vij ]is the matrix of V+
k. The rows of the simplicial matrix
[V]k+1 are the barycentric coordinates of the vertices vi∈ V+
k+1. It follows
that [V]k+1 is a non-negative matrix with the property that the sum of the
coordinates in each row is equal to 1. Alternatively, since v1∧ · ·· ∧ vk+1 6= 0,
and not requiring the coordinates to be barycentric, the matrix [V]k+1 becomes
the transition matrix from the basis of null vectors Ak+1 to the basis vectors
vi∈ V+
k+1, for which all the relations found after (27) remain valid.
The content of V+
k+1 is
v△k=∧k+1
i=2 (vi−v1) = (v2−v1)∧ ·· · ∧ (vk+1 −v1)6= 0,(55)
in the geometric algebra A+
k+1 of Rk+1. Similar to (52) and (53), we have
xv△k=x·v△k+x∧v△k,
but there is no obvious simplification as found for null simplices in (52).
4.1 LPGG Calculus of S+
n⊂Rn+1
I will now give a brief introduction to the general theory of LPGG Calculus.
Standard geometric calculus has been in continual development over the last half
Century [5, 6, 10, 17]. Every signed graph V±
nof n-vertices can be studied in
terms of any of the geometric algebras determined by the sequences of signs (59),
(61), found in Appendix B. I will limit my discussion here to signed positive 1
2-
graphs V+
min Rn+1, using the barycentric coordinates of the convex null symplex
S+
n:= S+
n(a1,...,an+1 )⊂ N,
and the geometric algebra A+
n+1 ≡G1,n. For n= 0, define
S+
0:= {a|a6= 0, a2= 0} ⊂ N
to be the single null vector a.
For m≤n, let v1,...vm+1 ∈Rn+1 denote the vertices of a signed simplex
V+
m:= V+
m(v1,...,vm+1 )⊂ S+
n,
where v1∧ · ·· ∧ vm+1 6= 0. Define a(n+1) ≡ {a}(n+1) by
{a}(n+1) := {a1,...,an+1 },
15
and by {∨
ai}(n), the set of ncorrelated null vectors obtained by leaving out ai,
{∨
ai}(n):= {a1,...∨
ai...,an+1 }.
When no confusion can arise, we shorten {a}(n)to a(n). For n= 3,
{∨
a1}(3) ={a2, a3},{∨
a2}(3) ={a1, a3},and {∨
a3}(3) ={a1, a2}.
Since the set of vectors {v}(m+1) are linearly independent,
∧v(m+1) := v1∧ · ·· ∧ vm+1 6= 0,
V+
mdefines an m-simplex with m+ 1 = m+ 1
m-faces. Each (m+ 1)-face is
geometrically represented by the oriented m-vector
∧∨
v(i):= v1∧ · ··∨
i··· ∧ vm+1.
Also, define the (m+ 1)-sum and the m-sum, by
Xv(m+1) := v1+···+vm+1 ,and Xv(∨
i):= v1+···∨
i···+vm+1 .
The signed complete graph V+
mis said to be closed if Piv(∨
i)= 0, and of order
k, if kis the largest number of linearly independent vertices of V+
m. Naturally,
we use the barycentric coordinates associated with S+
n, and, without loss of
generality, assume that the position vector x∈ V+
m+1, is given by
x= (x1,···, xm+1) :=
m+1
X
i=1
xiai∈Rm+1 ⊂Rn+1,
although other coordinate systems can be used.
I now restrict attention to studying the graph V+
m+1 of a particular m-
dimensional polytope. For x∈ V+
m+1, calculate
x2=m+1
X
i=1
xiai2
=X
0≤i<j≤m+1
xixj,
for all i, j,0< i 6=j≤m+ 1, and where the vertices of the m-polytope satisfy
v1∧ · ·· ∧ vm+1 6= 0.
Since V+
m⊂ S+
n, the barycentric coordinates xiof xwill all be positive, so
that
|x|2=x2=X
0≤i<j≤m+1
xixj≥0.(56)
For x∈ V+
m, define
|x|=sX
0≤i<j≤m+1
xixj≥0.
16
The points x∈ V+
m⊂ S+
n, for which |x|= 0, are exactly those points xof the
graph on the light cone. For all interior points of V+
m, where |x|>0, define the
unit vector
ˆx:= x
|x|.(57)
Since x∈ S+
nis barycentric, its coordinates satisfy Pxi= 1. Taking the
partial derivative ∂iof this equation, gives ∂iPn+1
j=1 xj= 0. By employing higher
order barycentric coordinates, based upon Hermite interpolation, this constraint
can be satisfied. Without going into details, for x∈ S+
n, I want to preserve the
property that ∂ix=aifor each 0< i ≤n+ 1, [14, 18]. The same effect can
be achieved by assuming, when differentiating x, we have relaxed the condition
that the coordinates of x∈Rn+1 are barycentric.
Recalling (26) and (30)
∇=2
nAn+1∂(n+1) −nˆ
∇=2
n∨
∇ − (n−1) ˆ
∇.(58)
For x∈Rn+1,∇x2= 2x,∇|x|= ˆx,∇|x|= ˆx, and ∇ˆx=n
|x|. These formulas
remain valid at all points x∈ S+
n, [6, p.66].
4.2 Platonic solids
Applying the decomposition formula (58),
∨
∇x=X
i
∨
ai
∂x
∂i =X
i
∨
aiai=n
2=n(n−1)
2,
which is the number of linear independent edges of S+
n.
The Laplacian ∨
∇2for the light cone projective geometry of S+
nis
∨
∇2=
n
X
i=1
∂2
i+n
2X
1≤i≤j≤n
∂i∂j.
Just as in Euclidean and pseudo-Euclidean geometry, the Laplacian ∨
∇2in S+
n,
is scalar valued.
For the signed simplex S+
3,
∨
∇2=∂2
1+∂2
2+∂2
3+∂2∂3+∂1∂3+∂1∂2.
For x∈ S+
n, we calculate
∨
∇2x2=
n
X
i=1
∂2
ix2+n
2X
1≤i≤j≤n
∂i∂jx2=n
22
.
I conclude with a Conjecture for n-Platonic Solids in (n+ 1)-dimensional
space Rn+1.
17
Conjecture: The number of n-Platonic Solids in any dimension nis equal
to the number of distinct n-Platonic Solids found in the n-simplex S+
n⊂Rn+1
with its vertices located at the null vectors a1,...,an+1 ∈ S+
n.
The number is known to be given by the sequence
{1,1,∞,5,6,3,3,3,···},
[19, 20, 21, 22, 23].
I want to welcome the reader to this beautiful new, but not really so new,
theory. Be careful - the calculations can be treacherous.
Acknowledgements
The seeds of this note were planted almost 40 years ago in discussions with Pro-
fessor Zbigniew Oziewicz, a distinguished colleague, about the fundamental role
played by duality in its many different guises in mathematics and physics [24].
The author thanks the Zbigniew Oziewicz Seminar on Fundamental Problems
in Physics group for many fruitful discussions of the ideas herein [25], and offers
special thanks to Timothy Havel for thoughtful comments about earlier versions
of this work. Not least, the author thanks the organizers of ICACGA2023 for
allowing me extra time to complete this work.
References
[1] Sobczyk, G.: Geometric Algebras of Compatible Null Vectors, IN-
TERNATIONAL CONFERENCE OF ADVANCED COMPUTA-
TIONAL APPLICATIONS OF GEOMETRIC ALGEBRA 2022.
http://dx.doi.org/10.13140/RG.2.2.27783.47521
[2] Dieudonné, J.D.: The Tragedy of Grassmann, Linear and Multilinear Alge-
bra, Vol. 8, pp. 1-14 (1979).
[3] W.K. Clifford, Applications of Grassmann’s extensive algebra, Am. J.
Math (ed.), Mathematical Papers by William Kingdon Clifford, pp. 397-401,
Macmillan, London (1882). (Reprinted by Chelsea, New York, 1968.)
[4] Sobczyk, G., Talk: Geometric Algebras of Compatible Null Vec-
tors, INTERNATIONAL CONFERENCE OF ADVANCED COMPU-
TATIONAL APPLICATIONS OF GEOMETRIC ALGEBRA, 2022,
https://www.researchgate.net/publication/364355245
[5] Hestenes, D., Sobczyk, G., Clifford Algebra to Geometric Calculus: A Uni-
fied Language for Mathematics and Physics, 2nd. Edition (1992), Springer
Nature Link: 978-94-009-6292-7.html
[6] Sobczyk, G.: New Foundations in Mathematics: The Geometric Concept of
Number, Birkhäuser, New York 2013.
18
[7] Sobczyk, G.: Periodic Table of Geometric Numbers, 12 March 2020.
https://arxiv.org/pdf/2003.07159.pdf
[8] Riesz, M., Clifford Numbers and Spinors, The Institute for Fluid Dynamics
and Applied Mathematics, Lecture Series No. 38, University of Maryland,
1958.
[9] Sobczyk, G.: Matrix Gateway to Geometric Algebra, Spacetime and Spinors,
Independent Publisher, Nov. 7, 2019.
[10] Lounesto, P.: Clifford Algebras and Spinors, Cambridge Univerity Press
(2001). https://users.aalto.fi/ ppuska/mirror/Lounesto/
[11] G. Sobczyk, Notes on Plücker’s relations in geometric algebra, Advances
in Mathematics, 363 (2020) 106959. An early version of this article is
https://arxiv.org/pdf/1809.09706.pdf
[12] Sobczyk, G.: Conformal Mappings in Geometric Algebra, Notices of the
AMS, Volume 59, Number 2, p.264-273, 2012.
[13] Doran, C., Hestenes, D., Sommen, F., Van Acker, N.: Lie groups as spin
groups, J. Math. Phys., Vol. 34, pp. 3642–3669 (1993).
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Theory, April 2001The American Mathematical Monthly 108(4) DOI:
10.2307/2695240. https://www.researchgate.net/publication/242251192
_The_Missing _Spectral_Basis_in_Algebra_and_Number_Theory
[15] Sobczyk, G.: Simplicial Calculus with Geometric Algebra, in Clifford Alge-
bras and Their Applications in Mathematical Physics, Edited by A. Micali, R.
Boudet, and J. Helmstetter, (Kluwer Academic Publishers, Dordrecht 1992).
http://geocalc.clas.asu.edu/pdf-preAdobe8/SIMP_CAL.pdf
[16] Wikipedia "Barycentric Coordinate Systems".
[17] Hitzer,E., Kamarianakis, M., Papagiannakis, G., Vasik,
P., Survey of New Applications of Geometric Algebra,
https://d197for5662m48.cloudfront.net/documents/publicationstatus/128130/
preprint_pdf/dfc488e3e3197f4bb6cb6636ab5c569b.pdf
[18] Langer, T., Seidel, H.P., Higher Order Barycentric Coordi-
nates, EUROGRAPHICS 2008/G. Drettakis and R. Scopigno
(Guest Editors), Vol 27 (2008), Number 2. https://domino.mpi-
inf.mpg.de/intranet/ag4/ag4publ.nsf/0/637fcbb7f3f5a70fc12573cc00458c99/
[19] Sobczyk, G., Light Cone Projective Graph Geometry, Unpublished
manuscript, November 16, 2021.
[20] Havel, T., An Extension of Heron’s Formula to Tetrahedra, and the Pro-
jective Nature of Its Zeros, https://arxiv.org/abs/2204.08089
19
[21] Khovanova, T., Clifford Algebras and Graphs,
https://arxiv.org/abs/0810.3322
[22] Baez, J., 2020, https://math.ucr.edu/home/baez/platonic.html
[23] Hestenes, D., Crystallographic space groups in geometric algebra, JMP,
2006. https://aip.scitation.org/doi/abs/10.1063/1.2426416
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Common, NATO ASI Series C: Mathematical and Physical Sciences Vol. 183
(1986).
[25] Cruz Guzman, J., Page, B., “Zbigniew Oziewicz Seminar on Funda-
mental Problems in Physics", FESC-Cuautitlan Izcalli UNAM, Mexico.
https://www.youtube.com/channel/UCBcXAdMO3q6JBNyvBBVLmQg
Appendix A: Geometric Algebra Identities in A+
1,n
Some basis identities of the geometric algebra
G1,n ≡ A+
1,n =Rn+1 := R(a1,...,an+1 ),
where ai·aj=1−δij
2.
1. x2=x1x2, x ·v1=1
2(x1v12 +x2v11), x ·v2=1
2(x2v22 +x2v21)
2. v1·v2=1
2(v11v22 +v12 v21), v1∧v2= det v11 v12
v21 v22 a1∧a2
3. (a1∧a2) = 1
2(a1−a1)∧(a1+a2) = 1
2f1e1,(a1∧a2)2=1
4,
4. For y=y1a1+y2a2,x∧y= det x1x2
y1y2a1∧a2
5. (x∧y)2= det y·x y2
x2x·y
Change of Basis Formulas for n+1=8
T8=
1
2
1
20 0 0 0 0 0
1
2−1
20 0 0 0 0 0
1 0 1 0 0 0 0 0
1 0 1
2
√3
20 0 0 0
1 0 1
2
1
2√3q2
30 0 0
1 0 1
2
1
2√3
1
2√6
√5
2√20 0
1 0 1
2
1
2√3
1
2√6
1
2√10 q3
50
1 0 1
2
1
2√3
1
2√6
1
2√10
1
2√15
√7
2√3
20
T−1
8=
11000000
1−1000000
−1−1100000
−1
√3−1
√3−1
√3−2
√30 0 0 0
−1
√6−1
√6−1
√6−1
√6q3
20 0 0
−1
√10 −1
√10 −1
√10 −1
√10 −1
√10 2q2
50 0
−1
√15 −1
√15 −1
√15 −1
√15 −1
√15 −1
√15 q5
30
−1
√21 −1
√21 −1
√21 −1
√21 −1
√21 −1
√21 −1
√21 2q3
7
Appendix B: Classification of Geometric Algebras
There is an extremely interesting relationship between plus and minus signs of
the squares of the standard basis elements of Gp,q, and the 8-fold periodicity
structure of Clifford geometric algebras. Consider the following:
1. {+},{−}, e1∈G1,0, f1∈G0,1Qsigns −
2. {++},{+−},{−−},Gp,q , p +q= 2 Qsigns −
3. {+++},{+ + −},{+− −},{− − −} p+q= 3, etc. Qsigns +
4. {+ + ++},{+++−},{+ + −−},{+− −−},{− − −−} Qsigns +
5. n+ 1
2=6
2= 15 Q(15) −
6. n+ 1
2=7
2= 21 Q(21) −
This obviously gives the infinite sequence
−−,++,−−,++,−−,+ + .... (59)
Real geometric algebras Gp,q are constructed by extending the real number sys-
tem Rby n=p+qanti-commuting vectors ei, fjwhich have sqares ±1, respec-
tively
Gp,q := R[e1,···, ep, f1,···fq],(60)
[5, 6]. A more concise treatment of this construction, and its relationship to
real and complex square matrices is [9].
Geometric algebras enjoy a very special 8-fold periodicity relationship [7]. A
basic understanding of this important periodicity relationship can be obtained
by studying the signs of the squares of the pseudoscalar elements for the geomet-
ric algebra Gp,q of successively higher dimensions. The ±signs over pseudoscalar
elements indicate the sign of the square of that element.
21
0. {a∈ N| a2= 0}. The null vector a6= 0 has the property that a2= 0.
1. {+
e1},{−
f1}:G1,0,G0,1;
2. {−
e1e2} { +
e1f1},{−
f1f2}:G2,0,G1,1,G0,2;
3. {−
e1e2e3},{+
e1e2f1},{−
e1f1f2},{+
f1f2f3}:G3,0,G2,1,G1,2,G0,3;
4. {+
e1e2e3e4},{−
e1e2e3f1},{+
e1e2f1f2},{−
e1f1f2f3},{+
f1f2f3f4}
5. {+
e1e2e3e4e5},{−
e1e2e3e4f1},{+
e1e2e3f1f2},{−
e1e2f1f2f3},
{+
e1f1f2f3f4},{−
f1f2f3f4f5}.
6. {−
e1e2e3e4e5e6},{+
e1e2e3e4e5f1},{−
e1e2e3e4f1f2},{+
e1e2e3f1f2f3},
{−
e1e2f1f2f3f4},{+
e1f1f2f3f4f5,{−
f1f1f2f3f4f5}.
This obviously gives the sequence,
+,−,−+−,−+−+,+−+−+,−+−+−+,··· (61)
The sequences (59) and (61) follow directly from the well known periodicity
laws of all real and complex geometric algebras [10]. The two sequences beau-
tifully reflect how any geometric algebra Gp,q , for n=p+qcan be represented
either as a real or complex matrix algebra of dimension 2n. In the case of the
complex matrix algebra, the imaginary number ican be interpreted as the pseu-
doscalar element e1f1···enfnfn+1 in the center of the real geometric algebra
Gn,n+1.
22
