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 [1] J. Kovacs, F. Fang, G. Sadler and K. Irwin, ” The Sum of Squares Law” arXiv : 1210.1446. [2] M. Hamermesh, Group Theory and its Applications to Physical Problems, (Addison-Wesley, Reading (1962). (Reprinted by Dover). ) W. Miller, Jr., Symmetry Groups and their Applications, (Academic Press, New York (1972) ) . J. F. Cornwell, Group Theory in Physics, (Three volumes), Volume 1, (Academic Press, New York (1997). ) [3] C. Castro and M. Pavsic, Progress in Physics vol 1 (April 2005) 31. Phys. Letts B 559 (2003) 74. Int. J. Theor. Phys 42 (2003) 1693. [4] R. Williams, The Geometrical Foundations of Natural Structure (Dover edition 1979). 9