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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,
2
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
