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Pythagoras’s Theorem in Seven Dimensions

Frederick David Tombe,

Northern Ireland, United Kingdom,

sirius184@hotmail.com

17th January 2018

Abstract. It will be argued that if Pythagoras’s Theorem can hold outside of three

dimensions, then the only possibility might be in the special case of seven dimensions,

but that even this would be highly doubtful.

Introduction

I. The Pythagorean trigonometric identity is a special case of Lagrange’s

Identity which only holds in three and seven dimensions. This fact is closely

tied up with the vector cross product. The purpose of this article is to show that

Pythagoras’s theorem only exists in three-dimensional space. The special case

of seven dimensions will be fully investigated as an illustration.

Pythagoras’s Theorem

II. The square of the hypotenuse of a right-angle triangle is equal to the sum of

the squares of the other two sides. This ancient rule in Euclidean geometry,

better known as Pythagoras’s Theorem, is actually only a special case of the

more general cosine rule,

c² = a² + b² − 2abcosθ (1)

as applied to cases where the angle θ is equal to ninety degrees. When θ is

ninety degrees, we will have a right-angle triangle with adjacent side length a,

opposite side b, and hypotenuse c. The concept of ‘angle’ relates to rotation in a

two-dimensional plane with the rotation axis being in the third dimension. If,

however a body were to be rotating in a two-dimensional plane in a four-

dimensional space, we would have to decide which of the other two dimensions

the axis of rotation would occupy. And if we have difficulty trying to imagine

rotation in a four-dimensional space, so also will we have difficulty trying to

imagine the cosine rule, and hence Pythagoras’s theorem in 4D.

It’s assumed however in pure mathematics and relativistic circles that

Pythagoras’s theorem can actually apply in spaces of any dimensions. In fact,

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on first examination, Pythagoras’s theorem does appear to hold in four

dimensions. The apparent legitimacy hinges on the operation of the “scalar

product” or “dot product” of two vectors, a∙b = ||a|| ||b||cosθ. If we define the

sides of a triangle using vectors, each expressed in terms of four mutually

orthogonal unit vectors, we do in fact find consistency with Pythagoras’s

theorem. A simple example is enough to demonstrate this. Consider two vectors

in a 4D space. Let vector a be defined as a = a1i + 0j + a3k + 0l and let vector b

be defined as b = 0i + b2j + 0k + b4l, where the four unit vectors i, j, k, and l,

are mutually orthogonal. Those components were specially chosen to make

vectors a and b orthogonal so that they can represent two sides of a right-angle

triangle in a hypothetical four-dimensional space. If we assume Pythagoras’s

theorem to be applicable in this four-dimensional space, then it will follow that

||a + b||2 should be equal to ||a||2 + ||b||2, and indeed this turns out to be the case,

||a + b||2 = a12 + b22 + a32 + b42 (2)

Meanwhile,

||a||2 + ||b||2 = (a12 + a32) + (b22 + b42) = a12 + b22 + a32 + b42 (3)

This is enough to convince relativists and pure mathematicians that

Pythagoras’s theorem can hold in spaces of any dimensions. But this application

of the scalar product only involved parallel and orthogonal vectors. There was

no evidence that it would hold more generally for the in-between angles if such

a concept as ‘angle’ could even be imagined in 4D.

The Minkowski Metric Tensor of Relativity

III. The metric tensor of special relativity has its roots in two consecutive

applications of Pythagoras’s Theorem. The first application is in respect of the

right-angle triangle that can be used to derive the time dilation formula of the

Lorentz transformations, while the second application relates to splitting one of

the sides of that triangle into three mutually orthogonal Cartesian components.

This results in a 4D space-time continuum, but since the Pythagorean

Trigonometric Identity cannot hold in 4D, the metric tensor cannot be said to

correspond to a 4D version of Pythagoras’s Theorem.

3

The Pythagorean Trigonometric Identity

IV. Consider the “Pythagorean Trigonometric Identity”,

sin2θ + cos2θ = 1 (4)

where the sine and cosine terms are derived from the squares of the two

perpendicular sides of a right-angle triangle. Re-arranging equation (4) and

multiplying across by ||a||2||b||2 we get,

||a||2||b||2(1 – cos2θ) = ||a||2||b||2sin2θ (5)

This introduces another binary operation known as the “vector product” or the

“cross product”. The right-hand side of equation (5) is the square of this vector

cross product. The cross product is defined as,

a × b = ||a|| ||b||sinθn

̂ (6)

where n

̂ is a unit vector mutually perpendicular to a and b. The underlying

principle behind the cross product is the idea that a bilinear map involving any

two elements in a set of three can yield the third. The original inspiration came

in 1843 when Sir William Rowan Hamilton was walking along the towpath by

the Royal Canal in Dublin. In a flash of genius, Hamilton discovered the three-

dimensional formula, i2 = j2 = k2 = ijk = −1, and in his excitement he cut it on a

stone of Brougham Bridge at Cabra. A plaque commemorating the occasion can

be seen at the location today. Hence equation (5) can be written in the form,

||a||2||b||2 − ||a∙b||2 = ||a × b||2 (7)

Equation (7) is known as the Lagrange Identity. The Lagrange identity itself

holds in any dimensions and it takes the more general form,

||a||2||b||2 − ||a∙b||2 = Σ (axby – aybx)2 (8)

where 1 ≤ x < y ≤ n, but it’s only in the case of three dimensions that it satisfies

the Pythagorean trigonometric identity. In the specific form expressed in

equation (7), it is impossible to write the Lagrange identify in any higher

dimensions, apart from one interesting exception, that being the case of seven

dimensions, which will be discussed in the next section.

4

The Special Case of Seven Dimensions

V. While it’s impossible to set up a bilinear vector cross product in an even

number of dimensions, it is nevertheless possible to set one up for any odd

number of dimensions. We can set up a table for five mutually orthogonal unit

vectors and map each unto two pairs from amongst the rest. We will not

however be able to fit the result into the Lagrange identity. And if we can’t fit it

with the Lagrange identity, then neither can we fit it with the Pythagorean

trigonometric identity. Only in the case of three and seven dimensions, ignoring

the trivial cases of zero and one, can we fit the result with the Lagrange identity.

This restriction follows from a nineteenth century theorem on composition

algebras by Adolf Hurwitz’s (1859 - 1919). More formal proofs only came

about relatively recently [1]. It will now be shown that the equality,

Σ (axby – aybx)2 = ||a × b||2 (9)

holds in 7 dimensions, where 1 ≤ x < y ≤ 7. We can set up a table of mutually

orthogonal unit vectors as follows,

i = j×l = m×n = k×o

j = k×m = n×o = l×i

k = l×n = o×i = m×j

l = m×o = i×j = n×k

m = n×i = j×k = o×l

n = o×j = k×l = i×m

o = i×k = l×m = j×n

Table 1.

Then if vector a = a1i + a2j + a3k + a4l + a5m + a6n + a7o, and vector b = b1i +

b2j + b3k + b4l + b5m + b6n + b70, the cross product is,

a×b = c1i + c2j + c3k + c4l + c5m + c6n + c7o (10)

where,

c1 = a2b4 – a4b2 + a5b6 – a6b5 + a3b7 – a7b3

c2 = a3b5 – a5b3 + a6b7 – a7b6 + a4b1 − a1b4

c3 = a4b6 – a6b4 + a7b1 – a1b7 + a5b2 – a2b5

c4 = a5b7 – a7b5 + a1b2 – a2b1 + a6b3 – a3b6

c5 = a6b1 – a1b6 + a2b3 – a3b2 + a7b4 – a4b7

5

c6 = a7b2 – a2b7 + a3b4 – a4b3 + a1b5 – a5b1

c7 = a1b3 – a3b1 + a4b5 – a5b4 + a2b6 – a6b2

Table 2.

In order to expand ||a × b||2 we therefore need to take the sum of the squares of

c1 through to c7 and this multiplies out to 252 terms. These 252 terms can be

split into two groups. There is a group of 84 terms, which can in turn be reduced

to 21 squared terms in brackets (shown in black below). It is this group of 21

alone that makes the equality between ||a × b||2 and Σ (axby – aybx)2. Then there

is a second group of 168 terms (shown in red below) which cancels out

completely. It is this cancellation, which ignoring the special case of 3

dimensions, is unique to 7 dimensions. It will not happen in 5 dimensions, and

neither will it happen in 9, 11, or any higher dimensions. The expansion is as

follows,

||a × b||2 = (a2b4 – a4b2)2 + (a5b6 – a6b5)2 + (a3b7 – a7b3)2 + 2(a2b4 –

a4b2)(a5b6 – a6b5) + 2(a2b4 – a4b2)(a3b7 – a7b3) + 2(a5b6 – a6b5)(a3b7 –

a7b3) + (a3b5 – a5b3)2 + (a6b7 – a7b6)2 + (a4b1 – a1b4)2 + 2(a3b5 –

a5b3)(a6b7 – a7b6) + 2(a3b5 – a5b3)(a4b1 – a1b4) + 2(a6b7 – a7b6)(a4b1 –

a1b4) + (a4b6 – a6b4)2 + (a7b1 – a1b7)2 + (a5b2 – a2b5)2 + 2(a4b6 –

a6b4)(a7b1 – a1b7) + 2(a4b6 – a6b4)(a5b2 – a2b5) + 2(a7b1 – a1b7)(a5b2 –

a2b5) + (a5b7 – a7b5)2 + (a1b2 – a2b1)2 + (a6b3 – a3b6)2 + 2(a5b7 –

a7b5)(a1b2 – a2b1) + 2(a5b7 – a7b5)(a6b3 – a3b6) + 2(a1b2 – a2b1)(a6b3 –

a3b6) + (a6b1 – a1b6)2 + (a2b3 – a3b2)2 + (a7b4 – a4b7)2 + 2(a6b1 –

a1b6)(a2b3 – a3b2) + 2(a6b1 – a1b6)(a7b4 – a4b7) + 2(a2b3 – a3b2)(a7b4 –

a4b7) + (a7b2 – a2b7)2 + (a3b4 – a4b3)2 + (a1b5 – a5b1)2 + 2(a7b2 –

a2b7)(a3b4 – a4b3) + 2(a7b2 – a2b7)(a1b5 – a5b1) + 2(a3b4 – a4b3)(a1b5 –

a5b1) + (a1b3 – a3b1)2 + (a4b5 – a5b4)2 + (a2b6 – a6b2)2 + 2(a1b3 –

a3b1)(a4b5 – a5b4) + 2(a1b3 – a3b1)(a2b6 – a6b2) + 2(a4b5 – a5b4)(a2b6 –

a6b2) (11)

Re-arranging the black subscripts numerically and segregating the black terms

from the red terms, which have now been expanded, we see more clearly how

the black terms are equal to Σ (axby – aybx)2, where 1 ≤ x < y ≤ 7. Hence,

||a × b||2 = (a1b2 – a2b1)2 + (a1b3 – a3b1)2 + (a1b4 – a4b1)2 + (a1b5 –

a5b1)2 + (a1b6 – a6b1)2 + (a1b7 – a7b1)2 + (a2b3 – a3b2)2 + (a2b4 – a4b2)2

+ (a2b5 – a5b2)2 + (a2b6 – a6b2)2 + (a2b7 – a7b2)2 + (a3b4 – a4b3)2 + (a3b5

6

– a5b3)2 + (a3b6 – a6b3)2 + (a3b7 – a7b3)2 + (a4b5 – a5b4)2 + (a4b6 – a6b4)2

+ (a4b7 – a7b4)2 + (a5b6 – a6b5)2 + (a5b7 – a7b5)2 + (a6b7 – a7b6)2

+2[a2b4a5b6 – a2b4a6b5 – a4b2a5b6 + a4b2a6b5

+ a2b4a3b7 – a2b4a7b3 – a4b2a3b7 + a4b2a7b3

+ a5b6a3b7 – a5b6a7b3 – a6b5a3b7 + a6b5a7b3

+ a3b5a6b7 – a3b5a7b6 – a5b3a6b7 + a5b3a7b6

+ a3b5a4b1 – a3b5a1b4 – a5b3a4b1 + a5b3a1b4

+ a6b7a4b1 – a6b7a1b4 – a7b6a4b1 + a7b6a1b4

+ a4b6a7b1 – a4b6a1b7 – a6b4a7b1 + a6b4a1b7

+ a4b6a5b2 – a4b6a2b5 – a6b4a5b2 + a6b4a2b5

+ a7b1a5b2 – a7b1a2b5 – a1b7a5b2 + a1b7a2b5

+ a5b7a1b2 – a5b7a2b1 – a7b5a1b2 + a7b5a2b1

+ a5b7a6b3 – a5b7a3b6 – a7b5a6b3 + a7b5a3b6

+ a1b2a6b3 – a1b2a3b6 – a2b1a6b3 + a2b1a3b6

+ a6b1a2b3 – a6b1a3b2 – a1b6a2b3 + a1b6a3b2

+ a6b1a7b4 – a6b1a4b7 – a1b6a7b4 + a1b6a4b7

+ a2b3a7b4 – a2b3a4b7 – a3b2a7b4 + a3b2a4b7

+ a7b2a3b4 – a7b2a4b3 – a2b7a3b4 + a2b7a4b3

+ a7b2a1b5 – a7b2a5b1 – a2b7a1b5 + a2b7a5b1

+ a3b4a1b5 – a3b4a5b1 – a4b3a1b5 + a4b3a5b1

+ a1b3a4b5 – a1b3a5b4 – a3b1a4b5 + a3b1a5b4

+ a1b3a2b6 – a1b3a6b2 – a3b1a2b6 + a3b1a6b2

+ a4b5a2b6 – a4b5a6b2 – a5b4a2b6 + a5b4a6b2] (12)

(In order to check that the 168 red terms exactly cancel, a larger version has

been prepared in Appendix I for the purposes of printing and marking off)

Despite this miraculous cancellation though, the fact that the seven-dimensional

vector cross product can’t easily be linked to rotation, casts serious doubt on the

idea that Pythagoras’s theorem could hold in seven dimensions. The Jacobi

Identity, a×(b×c) + b×(c×a) + c×(a×b) = 0, is closely associated with rotation,

but the fact that it doesn’t apply to the seven dimensional cross product further

casts doubt on the validity of Pythagoras’s Theorem in 7D.

The Inertial Path

VI. Consider a body in motion in an inertial frame of reference. We can write

the position vector of this body relative to any arbitrarily chosen polar origin as,

7

r = rr

̂ (13)

where the unit vector r

̂ is in the radial direction and where r is the radial

distance. Taking the time derivative and using the product rule, we obtain the

velocity,

= r

̂ + rω (14)

where is the unit vector in the transverse direction and where ω is the angular

speed about the polar origin. Taking the time derivative a second time, we

obtain the expression for acceleration in the inertial frame,

r

̈ = r

̈r

̂ + ω + ω + r(dω/dt) − rω2r

̂ (15)

Re-arranging and multiplying across by mass m leads to,

mr

̈ = m(r

̈ − rω2)r

̂ + m(2vrω + r(dω/dt)) (16)

†see the note at reference [3] regarding Maxwell’s equation (77)

where ω is the angular speed and vr is the radial speed. The radial component of

equation (16) contains a centrifugal force, mr

̈, and an inertial centripetal force,

−mrω2, while the transverse component contains a Coriolis force, mr(dω/dt),

which equals 2mvrω when angular momentum is conserved. In the case of

uniform straight-line motion, the total acceleration is zero, but when a constraint

is introduced, an imbalance occurs in the inertial symmetry. For example, if the

body is tethered to a pivot, the inertial centrifugal force pulls on the constraint,

hence inducing a reactive centripetal tension within the material of the

constraint. This tension cancels with the inertial centrifugal force and the

resultant is a net inertial centripetal force which curves the path of motion.

The inertial centripetal force −mrω2 in equation (16) with respect to one

polar origin, is an inertial centrifugal force with respect to the origin at the same

distance along a line through the moving body on the other side of it. From the

perspective of the moving body, there is therefore a centrifugal force to every

point in space giving rise to a cylindrical vector field in the likeness of the

magnetic field that surrounds an electric current. The centrifugal force to any

point on a particular cylindrical shell, concentric to the path of motion, will be a

resolution of the centrifugal force to a point on the shell, that acts

perpendicularly to the path of motion. The perpendicular centrifugal force will

drop off with an inverse cube law in distance from the moving body (see

equation (18)). Since centrifugal force is the radial gradient of kinetic energy, it

8

is now proposed that this cylindrical vector field represents the extension of the

body’s kinetic energy.

The idea that a moving entity could yield up energy to a surrounding

medium and have it returned during deceleration is observed in the case of an

electromagnetic field. When the power supply to an electric circuit is

disconnected, its magnetic field collapses and its stored energy, ½LI2, flows

back into the circuit giving the current a final surge forward. Another rather

obvious connection between the inertial forces and magnetism is the fact that

the Coriolis force has a similar form to the magnetic force, F = qv×B, if we

adopt Maxwell’s idea that it is caused by a sea of molecular vortices pressing

against each other with centrifugal force while striving to dilate [2], [3], [4], [5],

and where the vorticity, H = 2ω, represents the magnetic intensity, where ω is

the circumferential angular speed of the vortices and where B = µH.

It is therefore proposed that kinetic energy, ½mv2, is a pressure, and an

extended pressure field which drops off with an inverse cube law in distance,

and that it is induced by the fine-grained centrifugal force interaction between

the immediately surrounding vortices and the molecules of the moving body as

they shear past each other. These vortices will be the rotating electron-positron

dipoles introduced in section I, and they will form double helix vortex rings

around the moving body, centred on the line of motion, similar in principle to

smoke rings. To the front and rear of the motion, the vortices would therefore

have to be continually aligning and de-aligning, and the associated precession of

the vortices would be fully compatible with a Coriolis force acting equally and

oppositely at the front and the rear of the motion. This process would be

identical in principle to Maxwell’s explanation for Ampère’s Circuital Law. The

kinetic energy pressure field, or inertial field, that accompanies a moving body

is therefore in principle just a variation on the magnetic field theme. It is a weak

magnetic field.

The Inertial Frame of Reference

VII. The inertial frame of reference is a relatively recent concept, introduced

mainly in connection with Einstein’s theories of relativity and retrospectively

applied to Newtonian mechanics. Newton only ever considered the background

stars as the significant frame of reference [6]. As a proposition we’ll take the

inertial frame of reference to be Maxwell’s sea of molecular vortices with the

vortices being rotating electron-positron dipoles [7], [8]. There is supposed to be

no gravitational field in an inertial frame of reference, yet if we want to have

one in practice, we have little choice but to choose a region of the electron-

positron sea which is entrained within the gravitational field of a planet. This

way we can have an inertial frame of reference providing that we ignore the

gravitational force. This is fine therefore when solving problems where gravity

9

is negligible. If on the other hand we are dealing with planetary orbital problems

where two inertial frames of reference are shearing past each other while

generating centrifugal force at the interface, this changes the physical basis

upon which the inertial forces are induced, and the only physically significant

directions are radial and transverse. In planetary orbits, conservation of angular

momentum causes the total transverse term in equation (16) to vanish. This is

recognized in Kepler’s second law, which is the law of equal areas. Meanwhile

the gravity sinks distort the inertial mechanism. Gravitational tension has a

physically cancelling effect on the centrifugal pressure forces that are measured

relative to the gravitating centres. Writing the centrifugal term in the form +rω2,

the problem in the radial direction reduces to the scalar equation,

r

̈ = −k/r2 + rω2 (17)

where k is the gravitational constant. Taking l to be the angular momentum

constant equal to r2ω, we can write Leibniz’s equation in the form,

r

̈ = −k/r2 + l2/r3 (18)

Between the two planets, the inter-play between the gravitational inverse square

law attractive force, which is a tension, and the inverse cube law centrifugal

repulsive force, which is a pressure, involves two different power laws, and this

leads to stable orbits that are elliptical, circular, parabolic, or hyperbolic. And

since the gravitational tails on the far sides of the planets will undermine the

inertial centripetal mechanism, then centrifugal force and gravity are the only

real forces acting in the radial direction.

Conclusion

VIII. The connection between rotation, Pythagoras’s theorem, the cosine rule,

the inertial forces, and electromagnetism, along with the fact that the

Pythagorean trigonometric identity only holds in three dimensions, suggests

unequivocally that space is a three-dimensional construction stabilized on

cylindrical symmetry. It is proposed that space is densely packed with tiny

dipolar vortices in which the default alignment is double helix toroidal vortex

rings forming magnetic lines of force. These vortices are responsible for the

inertial forces, magnetic force, electromagnetic induction, and electromagnetic

radiation, and they also absorb the vorticity out of the large-scale gravitational

sinks. A vortex involves a rotation in a two-dimensional plane with the rotation

axis in the third dimension.

10

References

[1] Silagadze, Z. K., “Multi-dimensional vector product” Budker Institute of Nuclear Physics,

Novosibirsk, Russia, (2002)

file:///C:/Users/user/Downloads/seven%20dimensional%20cross%20product.pdf

[2] Clerk-Maxwell, J., “On Physical Lines of Force”, Philosophical Magazine, Volume

XXI, Fourth Series, London, (1861)

http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf

† Equation (77) in Maxwell’s paper is his electromotive force equation and it exhibits a

strong correspondence to equation (16) in this article. The transverse terms 2mvrω (where

vorticity H = 2ω) and mr(dω/dt) (where rω is the transverse speed) correspond respectively

to the compound centrifugal term µv×H and the Faraday term −∂A/∂t, with m corresponding

to µ, and where A is the electromagnetic momentum.

[3] Whittaker, E.T., “A History of the Theories of Aether and Electricity”, Chapter 4, pages

100-102, (1910)

“All space, according to the younger Bernoulli, is permeated by a fluid aether, containing

an immense number of excessively small whirlpools. The elasticity which the aether appears

to possess, and in virtue of which it is able to transmit vibrations, is really due to the

presence of these whirlpools; for, owing to centrifugal force, each whirlpool is continually

striving to dilate, and so presses against the neighbouring whirlpools.”

[4] O’Neill, John J., PRODIGAL GENIUS, Biography of Nikola Tesla, Long Island, New

York, 15th July 1944

http://www.rastko.rs/istorija/tesla/oniell-tesla.html

“Long ago he (mankind) recognized that all perceptible matter comes from a primary

substance, of a tenuity beyond conception and filling all space - the Akasha or luminiferous

ether - which is acted upon by the life-giving Prana or creative force, calling into existence,

in never ending cycles, all things and phenomena. The primary substance, thrown into

infinitesimal whirls of prodigious velocity, becomes gross matter; the force subsiding, the

motion ceases and matter disappears, reverting to the primary substance”.

[5] Lodge, Sir Oliver, “Ether (in physics)”, Encyclopaedia Britannica,

Fourteenth Edition, Volume 8, Pages 751-755, (1937)

http://gsjournal.net/Science-

Journals/Historical%20PapersMechanics%20/%20Electrodynamics/Download/4105

In relation to the speed of light, “The most probable surmise or guess at present is that the

ether is a perfectly incompressible continuous fluid, in a state of fine-grained vortex

motion, circulating with that same enormous speed. For it has been partly, though as yet

incompletely, shown that such a vortex fluid would transmit waves of the same general nature

as light waves— i.e., periodic disturbances across the line of propagation—and would

transmit them at a rate of the same order of magnitude as the vortex or circulation speed”

[6] Dingle, H., “On Inertia and Inertial Frames of Reference”, Quarterly Journal of the

Royal Astronomical Society, Volume 8, Page 262 (1967)

http://adsabs.harvard.edu/full/1967QJRAS...8..252D

[7] Tombe, F.D., “The Double Helix Theory of the Magnetic Field” (2006)

Galilean Electrodynamics, Volume 24, Number 2, p.34, (March/April 2013)

11

http://gsjournal.net/Science-Journals/Research%20Papers-

Mathematical%20Physics/Download/6371

[8] Tombe, F.D., “Induction of Electrostatic Repulsion by Strong Gravity” (2017)

http://gsjournal.net/Science-Journals/Research%20Papers-

Mechanics%20/%20Electrodynamics/Download/7167

Appendix I

Page 12 below can be printed out for the purpose of marking it off with a pen, in

order to demonstrate the total cancellation. These are 168 (2 × 84) of the 252

terms which resulted in equation (12) above when ||a × b||2 was expanded in

seven dimensions. This cancellation would not have worked in five dimensions,

nor will it work for any higher dimensions, and a product can’t even be

constructed in the first place for any even dimensions, which hence rules out the

case of four dimensions. While fifteen dimensions might have been the next

obvious one to try, based on the series 0, 1, 3, 7, 15, - - - - - (2n – 1), this would

mean dealing with 2,940 (14×14×15) terms as opposed to the 252 (6×6×7)

terms in this seven dimensional case, or the mere 12 (2×2×3) terms in the three

dimensional case. However, on knowing Adolf Hurwitz’s theorem on

composition algebras and Silagadze’s proof [1], there would be little point in

trying. The key below will assist with finding matching pairs on page 12.

Rows 1-21, columns A-D

(1A, 21C), (1B, 8D), (1C, 8A), (1D, 21B),

(2A, 16C), (2B, 15A), (2C, 15D), (2D, 16B),

(3A, 11B), (3B, 4D), (3C, 4A), (3D, 11C),

(4B, 11D), (4C, 11A), (5A,19C), (5B, 18A),

(5C, 18D), (5D, 19B), (6A, 14B), (6B, 7D),

(6C, 7A), (6D,14C), (7B,14D), (7C, 14A),

(8B, 21A), (8C, 21D), (9A, 17B), (9B, 10D),

(9C, 10A), (9D, 17C), (10B,17D), (10C, 17A),

(12A, 20B), (12B, 13D), (12C, 13A), (12D, 20C),

(13B, 20D), (13C, 20A), (15B, 16D), (15C, 16A),

(18B, 19D), (18C, 19A).

12

A B C D

+2[a2b4a5b6 – a2b4a6b5 – a4b2a5b6 + a4b2a6b5 1

+ a2b4a3b7 – a2b4a7b3 – a4b2a3b7 + a4b2a7b3 2

+ a5b6a3b7 – a5b6a7b3 – a6b5a3b7 + a6b5a7b3 3

+ a3b5a6b7 – a3b5a7b6 – a5b3a6b7 + a5b3a7b6 4

+ a3b5a4b1 – a3b5a1b4 – a5b3a4b1 + a5b3a1b4 5

+ a6b7a4b1 – a6b7a1b4 – a7b6a4b1 + a7b6a1b4 6

+ a4b6a7b1 – a4b6a1b7 – a6b4a7b1 + a6b4a1b7 7

+ a4b6a5b2 – a4b6a2b5 – a6b4a5b2 + a6b4a2b5 8

+ a7b1a5b2 – a7b1a2b5 – a1b7a5b2 + a1b7a2b5 9

+ a5b7a1b2 – a5b7a2b1 – a7b5a1b2 + a7b5a2b1 10

+ a5b7a6b3 – a5b7a3b6 – a7b5a6b3 + a7b5a3b6 11

+ a1b2a6b3 – a1b2a3b6 – a2b1a6b3 + a2b1a3b6 12

+ a6b1a2b3 – a6b1a3b2 – a1b6a2b3 + a1b6a3b2 13

+ a6b1a7b4 – a6b1a4b7 – a1b6a7b4 + a1b6a4b7 14

+ a2b3a7b4 – a2b3a4b7 – a3b2a7b4 + a3b2a4b7 15

+ a7b2a3b4 – a7b2a4b3 – a2b7a3b4 + a2b7a4b3 16

+ a7b2a1b5 – a7b2a5b1 – a2b7a1b5 + a2b7a5b1 17

+ a3b4a1b5 – a3b4a5b1 – a4b3a1b5 + a4b3a5b1 18

+ a1b3a4b5 – a1b3a5b4 – a3b1a4b5 + a3b1a5b4 19

+ a1b3a2b6 – a1b3a6b2 – a3b1a2b6 + a3b1a6b2 20

+ a4b5a2b6 – a4b5a6b2 – a5b4a2b6 + a5b4a6b2] 21

13

14