Hoops, Primal Maths, Compatible Coordinates, Glossary.

R.H.Beresford. 11th. October 2006.
"Make things as simple as possible, but no simpler." (Albert Einstein.)

1. Introduction.

This is a glossary for the concepts developed in GroupLoopHoop', and demonstrated in GroupLoopdemo, Hoopdemo, (in http://library.wolfram.com/infocenter/MathSource/4894/.) & Hoops (in http://library.wolfram.com/infocenter/MathSource/6198/)  i.e. "Primal" (non-negative) numbers, "generalized signs", "conservative" Moufang Loops, and "Hoops" - generalized "partial-fraction division algebras" with "multiplicative remainders" and conserved symmetries or "sizes". Some Hoops have "orbit-generating" polar forms leading to powers and roots. "Orbits" are multiphase sinusoids; their mathematics involves generalized signs. Noether's theorem suggests relationships between them and fundamental particles. Generalized signs lead to "compatible coordinates" (Section 3) which unify and simplify coordinate geometry. Section 4 is a conjecture about "mixed metrics" and Section 5 discusses Equation Synthesis, which is used to find non-linear differential equations with travelling pulse solutions.
Please send any comments to me at rhberesford@btinternet.com

2. Glossary.

a,b,..or (Subscript[a, 1]), (Subscript[a, 2]),  Elements representing Primal (non-negative) numbers, coefficients in a director.
A	        A Director, an indexed set {a,b…} of m primal elements; also written (∑_(i = 1)^m)(Subscript[a, i]) (Subscript[d, i]).
A	        A Vec, an indexed set {a,b…} of m/n signed elements; also written (∑_(i = 1)^(m/n))(Subscript[a, i]) (Subscript[x, i]). (Only takes
	             on the properties of a vector in some algebras).
\(TraditionalForm\`\(\(\ \)\(𝔸 = 
        0 ⋃ \(ℝ\^+\)\)\)\) The Primal (unsigned continuous, Absolute) numbers, the union of zero and the half line.
Additive elimination. The generalization of subtraction A+(-A)=0 to A+(Subscript[A, i])+(Subscript[A, j])+...=0. (Subscript[A, i]) etc are negations.
Ai= \(TraditionalForm\`∑\+j\((∑\+i\( η\_i\) ∂\_\(a\_\(\(i\)\(\ \)\)\)f\_\(\(j\)\(\ \)\))\)\) /(Subscript[f, j]). The multiplicative inverse of A. Calculated by genInverse. Ai.A={1,0,..}
Annihilator.	A non-trivial vector with all sizes zero. E.g.{1,1,1,1,-2,-2} in D3.
arcTanh[a,o] A generalization of Mathematica's ArcTanh[a/o] that gives a complex hyperangle depending
	           on the a,o (adjacent,opposite) octant.
binegation.  Ordinary (unary) negation -A, with -(-A)=A, resulting from negation by C2 equivalencing. 
	         Often incorrectly conflated with subtraction, which is a binary operation B-A. (cf. ternegation)
(Subscript[C, 2]), (Subscript[C, 3]) etc.,  The cyclic groups (associative loops) of 2,3, etc. primal elements. Do not confuse with Rings (Subscript[ℤ, 2])
   or C2, C3.   etc, for which negation is defined. Negation is undefined for loops or groups. 
C3C2 etc.    The direct composition of (Subscript[C, 3]) with (Subscript[C, 2]) etc. Indirect composition is shown as e.g. KiC4, C8pC2.
C3	       "Ternal", "Terplex"; the algebra of three elements, equivalence relationships on C3C2, C3C4 etc.
Cartesian-  A director or vec as an indexed set of coefficients with related directions or dimensions.
    Form.   Many algebras have dual Polar Forms (q.v.) with offsets, radii, and angles. See toPolar, toVector.
Collapse.  Used in earlier versions, now replaced by Fold.
Compatible-  The uniform nomenclature produced by putting A (≡(Subscript[a, i]) (Subscript[d, i])) = f[(Subscript[ϕ, i])] where (Subscript[ϕ, i]) is an angle
   Coordinates measured towards the direction (Subscript[d, i]), from the hyperplane (Subscript[a, i])=0. f[(Subscript[ϕ, i])] = Sin[(Subscript[ϕ, i])] gives a 
    Direction-   generalised spherical "direction Sines" algebra with an euclidean metric; f[(Subscript[ϕ, i])] = ((Sin[(Subscript[ϕ, i])])^2) gives
    Sines.	a city block metric. The i'th component vanishes when (Subscript[ϕ, i])=0. (Direction Cosines lack this
    		 automatic elimination.)
Conservative.The property (discovered by Frobenius for group inverse tables) of some multiplication tables
	         to conserve the determinant on multiplication. Possessed by "Hoop Division algebras" - many
	         signed tables, all groups, and some non-associative Moufang loops including Octonions. I found
	         (empirically) that direct table conjugate factors are conserved.
(Subscript[d, i])                A basis direction for a loop. (Subscript[x, i]) is a basis dimension, folded from 2 or more directions.
	       Directions can only be associated with operators or primal coefficients.
Director.   See A. Directors are the objects multiplied by loops. If the loop has r-fold symmetry, a director
	     may fold to give a vec with m/r signed elements & dimensions, operated on by a hoop algebra.
Dozal, Doz. The algebra of 12-element directors and vecs, and the unital algebra derived from it.
Dual.	        Polar description (if one exists) for a vec as a set of offsets, squared radii or ulnae, and angles.
(Subscript[D, 3]),(Subscript[D, 4]), etc.  Dihedral groups, symmetries of the triangle, square, etc. Written D3, D4 when used as hoops.
Element.  (Subscript[a, 1])(Subscript[d, 1]), (Subscript[b, 2])(Subscript[x, 2]), c etc. A constituent part of a director or vec, consisting of a coefficient ((Subscript[a, 1]), (Subscript[b, 2])) and a
	     direction or dimension ((Subscript[d, 1]),(Subscript[x, 2])) (often implied by the index or the position of the element in the set).
Equation Synthesis. A method of finding non-linear differential equations with specific solutions by setting
	      up a conservative DE with free parameters, and an expression (typically for a compact travelling
	      wave) with undetermined coefficients. Substituting the expression & its derivatives into the
	      equation leads to constraints between the coefficients and parameters, thereby defining a specific
	      function that satisfies the specific equation. See Section 5. for examples.
Equivalencing.  The creation of a signed hoop (or of a vector) by "folding", an equivalence relationship
	     between sets (pairs, triples, quads etc) of primal elements (Subscript[r, i])~((Subscript[a, i]) - (Subscript[a, i + m/r])) (Subscript[x, i]). Impossible for
  	     some loops, such as (Subscript[C, 3]) & (Subscript[D, 3]), which may, however, be folded after first composing with (Subscript[C, 2]).
	     Complex hoops and vectors are folded from quads of primal elements, i.e. r=4. Generalized or 
	     extended signs are created by folding. (See Fold.)
Extended Signs. Generalizations of +,-,+i,-i; E.g. (𝕤^𝕣), (𝕕^12), (𝕟^9), (𝕠^8), (𝕘^7), (𝕙^6), (𝕡^5), (𝕀^4), (𝕐^3), (𝕁^3), (𝕜^2), (𝕝^2), (𝕞^2)=+1
(Subscript[f, j])	  The j'th distinct factor of the determinant of G. It is conserved, if G is a hoop, on multiplication;
	      if it is linear (L1) it is also conserved on addition. Every factor is a "size" for the hoop algebra.
Fold.  See Equivalencing, signed table. The conversion of a loop with m unsigned elements to a hoop with
	    m/r signed elements, often generating a signed multiplication table. Groups can operate as algebras
	    over fields because any group X can be considered to be folded from XF, where F is a field
	    group such as C2 (real) or C4 (complex). Vecs (generalized vectors) are folded from unsigned sets. 
Fragment. A linear term in a compact size expression e.g. (a-b) in ((((a - b))^2)+(((b - c))^2)+(((c - a))^2))/2; also the
	     quadratic terms in compact expressions for quartic sizes.
Frobenius conservation. The propery that Det[A] Det[B]=±Det[AB] when vectors are multiplied using a
	   Cayley Table. Det is the conjugate of that of the Cayley table, mapped with the vector elements.
	   Frobenius proved that all associative tables (groups) are conservative. I found that Octonions and
	   some other non-associative Moufang loops are conservative, and the property survives on
	   folding to a signed table.
Gaussian Pulse.  Exp[go + i ω T+ (∑_(x, y,  .. ))[i kx X - jx²(X+xi-vx T)²]. See Section 5.
G:(Subscript[d, i])×(Subscript[d, j])→ (Subscript[d, ij]) A finite loop, group, hoop or groupoid giving the direction or basis dimension (Subscript[d, ij]) resulting
	   from the multiplication of two directions (Subscript[d, i]), (Subscript[d, j]). Det[G] is that of the symbolic Cayley Table.
Generalized Conjugate. An involute vec that gives the conserved determinant factors. Real vecs are their
	own conjugate. The terplex conjugate of {a,b,c} is {a,c,b}. Th complex conjgate of {r,I} is {r,-i}.
Generalized Negation. Equivalence relations on sets of r primal coefficients (Subscript[e, i])~{(Subscript[a, i]), (Subscript[a, i + m/r]),  ... , 
            (Subscript[a, i + (r - 1) m/r])}.
Generalized Signs. Generalizations of +, -, +i, -i; with (𝕤^𝕣)=+1. Extended Signs gives a proposed standard.
hoopInverse[A,hoop] The hoop multiplicative inverse Ai, permitting division in the specified hoop; Ai·AB=B.
	  Calculated as sums of derivatives of shapes wrt "specific variables" that are fixed by the location of
	  the 1's in the index table. In signed tables, the terms have to be multiplied by the sign of each 1
	  The location and multiplier are supplied as gi in the shape of the hoop. This is a Cramer's rule
	  calculation, so the inverse has the determinant as a divisor. When this factorises, the inverse splits
	  into partial-fractions with each size becoming a denominator. Division is left multiplication by an
	  inverse. Multiplication and division both eliminate infinite partial-fractions (created by zero sizes)
	  by "ejecting" remainders, giving a renormalised result matching a pseudo-inverse. The result is
	  "projected" onto a constrained sub-algebra. 
hoopPower[A,power]  A procedure to convert a vec to a power or root. It raises offsets and radii to the
	   required power, and multiplies angles by the power. The target hoop polar form must be known.
	   If angles exceed 2π they "wrap round", and then ((((A^p)))^(1/p)) will be a rotated version of A.
hoopTest[table]  A procedure that reports whether the table is conservative and has division, powers, etc. 
hoopTimes[A,B,hoop]   The hoop multiplication rule, multiplying vecs A & B according to the specified
	   hoop. If either multiplicand has sizes that become zeroe in the product, these are "ejected" into
	   "remainders" so that size conservation is maintained. The product is "projected" into a sub-algebra.
gi,gp       Lists created by sh, used by genInverse & toPolar. 
gmmnn   A mnemonic for groups given by GAP>SmallGroup(mm,nn), or loops in the extended GAP Atlas.
Hoop Algebra. Multiplication and Division of vecs using a hoop multiplication table, with addition and
	    generalized subtraction that may use generalized signs.
Helix Identity.  (∑_(n = 1)^(n = m))Sin[t + 2  n π/m]^(2  p)=m Binomial[2p,p]/(4^p) (p<m/2 if m is even, else p<m)
	   The generalisation of  (Sin^2)[x] + (Cos^2)[x] = 1.  Derived by J.J.Thwaites (personal communication).
Hexal, Hex. The algebra of 6-element directors and vecs, and the unitary algebra derived from it.
Heyting Algebras use intuitionistic (not Boolean) logic. The middle is not excluded; ¬¬A is the interior of A.
Hyperbolic,  The analog of "polar" where a size includes a difference of 2 squared fragments. The Klein
    Hypolar.     group is the archetype, with squared ulnae (a±c)^2-(b±d)^2 and angles arcTanh[a±c,b±d].
𝕙,𝕀,𝕁,𝕜,𝕝 etc.  Generalized signs, different roots of unity. Proposed usage in Extended Signs. 'h etc. in ASCII.
i	        ((+1)^(1/4)), usually written ((-1)^(1/2)), a complex unit element. Ambiguous in Dav & C4C4. etc.
Incantation. My name for the expressions used to create a Cayley table using any of the procedures ca, cd,
	         cl, co, ge, ma, md, ms, pe, or ts.
Isomorphs, Isotopes I define Isomorphs as loops with renamed elements whilst isotopes are loops
	    with some changed signs that conserve the structure, but not the details, of the shape.
Hoop.	      An algebra (H,+,·,\,/,1) with addition +, Moufang ·, left \ & right / division, 1,  generalized signs,
	       and the Frobenius determinant conservation property. All groups, a few non-associative
	       Moufang Loops, and some signed tables folded from both, are Hoops. They are in the database
	       as "Protoloop" preferred Cayley table isomorphs. Loop names ending in n are not hoops.
Involutions. See Plex-conjugates. Vecs with changed signs of some elements. [Lounesto p29]
Indexed set. A set in which an index identifies each element, so that permuted sets are set isomorphs.
K	       The Klein four-group, and the corresponding algebra, which has three square roots of +1, and
	         equivalences to a "double algebra" with two. See Hyperbolic, Study numbers.
KdVB	     Korteweg-deVries broadened equation. [ n(1 + n r) (A^(1 + 2/n)) (Subscript[A, x])+(r - 1) ((Subscript[A, x])^2)/A(Subscript[( +(Subscript[A, x x])]), x]) +
	       (Subscript[([(A^(1 + m)) (n^2)(
                      r + m)]),  t]) =0. Some multi-dimensional cases have travelling pulse solutions. Section 5.2.
Large number  The observation that several ratios in physics are (approximately) low powers of (10^40); 
  hypothesis.      expressed by Dirac as (e^2)≃(10^40)(Subscript[G, m])t
L1, L2, etc. Linear (L1), quadratic (L2) etc. sizes, conserved in an algebra, and given by shape.  L26ζλϕ is
	       the quadratic (L2) size which splits into six squared fragments and contains ζ, λ, & ϕ in its polar
	       form. L1 sizes are conserved on addition, all sizes are conserved on multiplication for hoops.
	       Originally misnamed "lengths"; the n'th root of Ln may be a length, and may provide a metric.
Length.     A constituent of a Polar, corresponding to either a scalar or, if the n'th root of Ln, to a radius or
	        ulna in polar or hypolar coordinates. (The other constituents are angles.)
Limiting   The non-linear partial differential equation that has any fixed-shape fixed speed wave, travelling
  Velocity    in any direction in d spatial and 1 time direction, as a solution. Massive and light-like particles arise
  Equation.  if pairs or directions combine to give dimensions, or if some directions are "rolled-up" (Kaluza
	       -Klein). Gtt-∑[Gxx]=0 (with Gxx=(d^2)G/d(X^2) etc. and summation is over x1...xd). See Section 5.1
Loop.	   An m×m quasigroup Cayley table with a 1, for a set with m elements. See Moufang Loop.
	    Protoloops include signed tables as well as strict (unsigned) loops.
m	   The number of primal elements in a loop, group or director. Hoops have m/r elements.
Moufang Loop. An m×m multiplication table for a set with m elements with L & R inverses (Subscript[L, i])(Subscript[R, i])=(Subscript[R, i])(Subscript[L, i]) and
	      the Moufang properties, including zx.yz=(z.xy)z, relaxed to "up to a sign" for signed tables. All
	      groups are Moufang Loops. A few non-associative Moufang Loops (e.g. octonions) are hoops. 
Multi-phase   (∑_(n = 1)^m) 
              c Cos[ωτ + 2  n k π/m + ϕ] (Subscript[ζ, n])+f[t], a solution to (Subscript[y, n])" = (a^2)((Subscript[y, n - 1])-2(Subscript[y, n])+(Subscript[y, n + 1]))+f"[t] or 
   Sinusoid.	(Subscript[y, n])'= a((Subscript[y, n - 1])-(Subscript[y, n + 1]))+f'[t]. ((Subscript[e, r])^(𝕤 x))generalizes (e^(i x))to define them.
n-potent. A vec which is its own n'th power in a particular algebra, (A^n)=A. An Idempotent if n=2.
OC311	    A typical continuous orbit, being the C3 orbit with both sizes equal to 1.
Offset.     The L1 sizes after conversion to polar form. See Radius, Polar Form. Symbols α, β, γ, δ.
Orbit.	    A unital parametric representation of a director or vec, with multi-phase sinusoidal elements
	     depending on at least one free frequency parameter {σ,τ..}. Greek letters α…κ are used as the
	     size parameters. An example is given under "Size". Putting τ = -((Subscript[E, p]) t-P.x)/ℏ  gives the orbit the
	     properties of a multi-phase deBroglie wave. (Related to the "discrete orbits" in a group.)
Pauli-σ Algebra.The algebra defined by the Pauli-σ 2×2 matrix multiplication table, conserving (t^2)-(x^2)-(y^2)-(z^2)
Phase.	   One or more cyclically varying element in an orbit. Each phase has a phase angle and a radius or ulna.
Plex-Conjugate. The generalization of complex-conjugate to some hoops, a vec with some elements negated or
	     (in ternary algebras) with 𝕁↔(𝕁^2) exchanges. Elements of the product of the vec and the conjugates 
	     (which are involutions) are related to the sizes of the vec, often as signed sums of squared elements.
	    The hoop generalised complex-conjugate plex is supplied (if known) in shape.  
Polar Form.  A Dual representation of a vec as scalars, radii and angles, generalising ordinary polar coordinates.
Polar Angle. An angle parameter in a polar form or orbit. Symbols σ, τ, ϕ, φ, χ, ψ.
Polymorphs. Quartic size with several fragmentations and plex (involution) formulations.
PolyHelix Identity.  (∑_(i = 0)^(i = m - 1))(((Sin[t + 2  iπ/m]))^(2  p))= (4^(-p))m (({{2  p}, {p}})), {p<m if m is even, else p<2m}.(John Thwaites,
               private comminication).
PowerSinusoid.   v=aSin[ωt + ϕ]^p, which is a solution to (Overscript[v, ¨]) +(1/p - 1) ((\(v\& . \) )^2)/v+p (ω^2)v=0. The travelling
	        wave V=(a Sin[(∑_(x, y,  .. ))kx*(x+xi-jx*t)])^p is a solution to Vtt+(1/p-1) (Vt^2)/V-pV(∑_(x, y,  .. ))(((jx kx))^2).
Primal,  𝔸.  Non-negative continuous. Real and Complex numbers are equivalence-relationships on pairs or
	       quads of primal numbers. Other equivalence relations on r primals give Terplex (r=3) etc.
Protoloop.  A prefered isomorph of a table, showing any structure; shapes are defined for the protoloop. The 
	    makeProtoloop (mp) procedure generates the Cayley table (with options to create some isomorphs).
Pseudo-root.    Non-abelian polar duals involve non-conserved size parameters and angle that do not add on
	   multiplication, but which have valid reversions to the cartesian form. When they are used to calculate
	   roots and powers, arbitrary angles and uncertainty are introduced, giving rotated "pseudo-roots".
Radius.   The length associated with an angle in a polar representation. Symbols ϵ, ζ, η, κ, λ. See Ulna.
Rational Pulse. v=\(TraditionalForm\`\((a/\((1 + ∑\_\(x, y,  .. \)\(jx(xi + x - kx\ t)\)^2)\))\)\^n\) See Section 5.
𝕤 	 The generalised sign or r'th root of unity, (𝕤^r)=1, often incorrectly treated as a point on the complex
	     unit circle, but best considered as a "direction". I write it as  's in printable ASCII.
Remainder. Multiplication (or division) by vectors with zero-sizes create left and right "remainders" with the
	   sizes that are present in the multiplicands but lost from the product, ensuring that hoops are
	   conservative and "division by zero" is eliminated.
Size.	The value of a single conserved property of a director or orbit, for a given hoop algebra. Thus the orbit
	   OC3αγ={α+2γ Cos[τ],α+2γ Cos[τ+2π/3 ],α+2γ Cos[τ+4 π/3]}/3 has individual sizes L1α=α, 
	   L23ζ2=(ζ^2), and so has a shape {α,(ζ^2)}, in the algebra C3; (hyper)complex-conjugate "Eigenvector
	   pairs" are sizes factorized into complex pairs (etc) but are not conserved. Sizes are always
	   symmetrical ideals.
Shape.   All the sizes of a director or orbit, found by shape[vec,mnemonic]; needed by genInverse, toPol etc.
sh[mnemonic]  The symbolic factors of the named loop, with gi for use with genInverse and gp for use with
	  toPolar, toVector. It also defines plex for many small algebras.
shape[{a_,b_...}]  Factor (size) data for the target hoop, supplying the shape of {a,b,..} as a list of individual
	   sizes {(Subscript[f, 1]),…(Subscript[f, k])} & setting gi, gp for genInverse, toPolar. It also sets plex for many small algebras.
Signed Table. A multiplication table for a set which forms a group or loop on appending its signed elements.
	The multiplication table is the upper left quarter (ninth, sixteenth, etc.) of the full Cayley table with
	 (possibly signed) elements folded, by equivalence relationships, from the full table elements.
Spin. 	A property of an orbit, which is reversed by reversing the sequence of the phases. Clockwise spin 
	   (Cos, -Sin, -Cos, Sin) is taken to be positive or Up.
Splitting.  The primal vector equivalent of subtraction, Split[A,B]=C if B+C=A.
Study Numbers.  a+𝕜 b, (𝕜^2)=1, folded K, the C2 algebra of the hyperbolic plane. Louenesto 2001 p24.
SymDiff.  Unsigned difference between numbers s=a↔b : s+Min[a,b]=Max[a,b]. Primal "additive inverse".
toPolar[v,h].    The procedures to convert cartesian representations v into polar forms using hoop h.
toVector[p,h].	The procedures to convert polar representations p into cartesian forms using hoop h.
Ternegation.	-(-(-A))=A in systems with ternary symmetry, instead of -(-A)=A in simple (bi)negation.
Terplex.  The algebra of the C3 table over the real or complex field.
Tetral.    The algebra of the C4 and K tables over the real or complex field.
Triality.  "sends a simple rotation to a positive isoclinic rotation and a positive isoclinic rotation to a negative
             isoclinic rotation". The Moufang identity results in a special case of Cartan's triality. Louenesto p309.
             See "Loops" in the GAP package.              
Ulna.	The hyperbolic analogue of radius. (u^2)=(x^2)-(y^2).
Unity.  The object {1,0,…,0}  in unitary algebras, which leaves any object unchanged after multiplication.
	 Not to be confused with the identity matrix. The product of a vec and its inverse is unity.
Unit.    An object (e.g. {2/3,-1/3,-1/3} in C3) having all sizes 0 or 1 in a particular algebra.
Unital.	A vec with sizes that multiply to 1 in a specific algebra, ignoring zeroes, is unital. Orbits are unital.
	 (Clashes with "unital algebras", which are monoids and include all hoops)
Univector. My name for the generators of Cayley-Dickson & Clifford algebras. The unit is scalar, products
	   are bivectors, trivectors, etc. Commonly and confusingly called vectors or scalars.
Vec.	  An ordered list of m primal , m/2 real , m/3 ternary or m/4 complex elements, etc, a director or
	   folded director. Only a Vector in some algebras.
α,β,…λ  Size parameters in an orbit, also the corresponding length parameters in a Polar Dual.
(Subscript[x, i])	  The i'th basis dimension. Two or more directions may fold to a real dimension (Subscript[x, i])~{(Subscript[d, i]),(Subscript[d, i + m/2])}. 
σ,τ,…,ψ  A frequency parameter in an orbit. Additive in abelian multiplication.
	  Also an angle parameter in a Polar.

3. Compatible Coordinates.

Complex, Polar, Spherical, and Cylindrical coordinates are incompatible - they use different measures for different coordinates, and so are not naturally extensible to higher spaces. Also, they contain an implicit Euclidean Metric,  length=Sqrt[x^2+y^2+...]

The use of directors with non-negative co-ordinates "(Subscript[a, i])" and orthonormal basis directors "(Subscript[d, i])" provides a standard formalism for any type of co-ordinate system. In place of the unhelpful variety of
({{({{Real ,  a }}), ({{Complex, \(\(\ \)\(a\  + \ \(\(i\)\(\ \)\(b\)\(\ \)\)\)\)}}), ({{Polar, \(\({r\ Sin[Theta], 
\ \ r\ Cos[
                            Theta]}\)\(\ \)\)}})}, {({{Spherical, \(\({\ \ r\ Sin[Theta]\ Cos[
                            Phi], \ \ 
r\ Sin[Theta]\ Sin[
                            Phi], 
\ \ \ \ \ r\ Cos[
                            Theta]}\)\(\ \ \ \ \ \)\)}}), ({{Cylindrical, \(\({r\ Sin[Theta]\ Cos[
                            Phi], 
\ \ r\ Cos[Theta]
\ \ \ h}\)\(\ \ \)\)}}),  }})
a director co-ordinate system is written (using the summation convention for repeated indices) in one of the one of the forms
          A = (Subscript[d, i])f[(Subscript[a, i])];         or       A = (Subscript[d, i])f[(Subscript[a, i])/(Subscript[A, i])],	 (i = 1.  ..  m)

where "(Subscript[A, i])" is the maximum for an "(Subscript[a, i])" with an upper bound. "(Subscript[a, i])" have lower bounds of zero. Note that the subscript is modular m; subscript [m+i] is identical to subscript [i]. 
	The "a"s are coefficients; they obey the rules of primal unsigned algebra, with addition, multiplication, and trichotomous order. Subtraction is "zero lower-bounded". They are measurements "towards" their direction, increasing as the point moves away from the hyperplane (Subscript[d, i])=0, unlike the Cos terms in polar, spherical, and cylindrical coordinates - which decrease towards their direction and give rise to negative derivatives.
	The (Subscript[d, i])'s are "directions" or "basis directors"; they generalize such symbols as "-", "i" and "-i", and obey a hoop product rule (Subscript[d, i]) (Subscript[d, i])= (Subscript[d, i j]). They are  implicitly present when numbers are written as indexed sets of coefficients. Thus {a1,a2} is another way of writing (Subscript[d, 1])(Subscript[a, 1])+ (Subscript[d, 2])(Subscript[a, 2]). "A" is a "director" and has "sizes" (related to factors of the hoop determinant) that are conserved on multiplication and (iff linear) on addition or subtraction. Division involves partial fractions with each size being a denominator. Multiplication or division by a director with a zero size makes that size zero in the result - it is projection onto the sub-space in which that size is constrained to be zero.
	The f's are functions that define the geometry The identity function give plane geometry, the Sin function gives spherical geometry, Fig 2:-

the (Sin^2) function gives City-block geometry, Fig 3. Mixed geometries, such as cylindrical geometry, use different f's for different directions.

4. Mixed Metrics.

Conjecture! If Time is a direction rather than a dimension, it may not combine with space directions via a Pythagorean or Minkowski metric. An interval between two events {(Subscript[t, 1]), (Subscript[x, 1]), (Subscript[y, 1]), (Subscript[z, 1])} and {(Subscript[t, 2]), (Subscript[x, 2]), (Subscript[y, 2]), (Subscript[z, 2])} might have a pythagorean space metric combined by a city-blocks relation for the time interval, (Subscript[s, 1, 2])=((Subscript[t, 1])-(Subscript[t, 2]))-\(TraditionalForm\`\@\(\((x\_1 - x\_2)\)\^2 + \((y\_1 - y\_2)\)\^2 + \((z\_1 - z\_2)\)\^2\)\).

4.1 6-Vector Light Cone?

On 7/11/5 I realised that the quartic size "L4" of C3C4c (and possibly of various related algebras) might define a light cone because it conserved the sum of six squared terms, minus a (dependent) seventh squared term.This could be a "local time", unlike the 4-Vector light cone, which has an independent time coordinate. L4 is non-negative, and Robert Israel ["6-Vector Light Cone?", sci.math, 4/1/6.] expressed it as two factors, each the sum of two squared terms.These areincluded as "l22a" & "l22b" in the C3C4 shape and in sh12.

5. Seeking Particle Equations via Equation Synthesis.

5.1 Equation Synthesis, travelling pulse.

I developed Equation Synthesis in an attempt to find solitonic non-linear differential equations that describe particles. Conservative evolution equations are set up with free parameters, as are target functions (usually travelling pulses of fixed shape) with undetermined coefficients. Substituting the function derivatives into the equation gives long expressions. Simplifying relationships between the free parameters and the coefficients may then lead to simple equations that are solved by specific functions.
	The technique is to define a pulse function - e.g. the rational pulse V=\(TraditionalForm\`\((a/\((1 + ∑\_\(x, y,  .. \)\(jx(xi + x - kx\ t)\)^2)\))\)\^n\), the Radial Sech pulse a Exp[1 - (Sqrt[1 + (∑_(x, y,  .. ))(x^2)])], the modulated Gaussian pulse Exp[go + i ω T+ (∑_(x, y,  .. ))[i kx X - jx²(X+xi-vx T)²], the power-sinusoid pulse a Sin[X + xi - vx T]^p, repeating lattice pulses such as the JacobiCN pulse - and to find the differentials Vt(=dv/dt), Vx(=dv/dx), Vxx, etc. (This shorthand for derivatives is widely used in the non-linear equation literature.) Various combinations of differentials are now calculated, usually with Vt in the denominator. This allows evolution equations  F( V, Vx,Vxx, etc)=M Vt to be handled easily (necessary with Mathematica 2 or 3; Mathematica 5 has a greatly improved Simplify operation). The sum of several combinations is calculated; then constraints which simplify the result are sought between the parameters. The simplified results are RHS multiplying factors M. Each simplification gives an equation which has a specific case of V as a solution.

Fig 4 shows the decay of various TX pulses with similar peaks. The choice of the Cos[x/3.6]^12 power-sinusoid is arbitrary; higher powers give closer approximations to the Gaussian pulse with longer repeat distances, just as JacobiCN pulses approach the Sech pulse.

esr = 
        Plot[{1/(1 + (x^2)/2), Cos[x/3.6]^12,  Exp[1 - (Sqrt[1 + (x^2)])], 
            Sech[x], (E^(-(x^2)/2))}, {x, -1, 6}, 
          PlotRange -> {0, 1}, 
          DisplayFunction -> Identity];
\(\(Show[esr, 
        Graphics[{Text[\ \*""(Cos^12)     Sech    Rsech"", {3.6,  .03}], Text[\ "Rational", {2.9,  .3}], Text[\ "Exp", {1.3,  .3}]}], 
        DisplayFunction -> $DisplayFunction];\)\)

Fig. 4. Rational, Rsech, Sech, (Cos^12), and Exp functions plotted as TX pulses..

Firstly the Limiting Velocity Equation is shown to describe disturbances travelling at a fixed speed in any number of space-like directions. Then a generalization of the Korteweg-deVries (KdV) equation is discussed. It has some multi-dimensional solutions. A modulated Gaussian pulse and a power sinusoid wave are investigated.

5.2 Limiting Velocity Wave Equation.

The general PDE for disturbances travelling at unit velocity in d+1 directions is the Limiting Velocity Wave Equation 
                                                    Gtt-∑[Gxx]=0 .
(with Gxx=(d^2)G/d(x^2) etc. and summation is over (Subscript[x, 1])...(Subscript[x, m])). It is valid for any unit velocity function G[t-Σx lx] where the lx are direction sines (or cosines). The equation can be written in the conservative form \(∂\_{t, 2}\)g[t,x] = (∑_x)\(∂\_{x, 2}\)g[t,x] .The TXYZ case is shown:-

\(Simplify[∂\_{t, 2}G[\(-t\) + x\ lx + y\ ly + 
              z\ lz] - 
∂\_{x, 2}G[\(-t\) + 
              x\ lx + y\ ly + 
              z\ lz] - 
\ ∂\_{y, 2}G[\(-t\) + 
              x\ lx + y\ ly + z\ lz] - ∂\_{z, 2}G[\(-t\) + x\ lx + 
              y\ ly + z\ lz]]\)

In each case there is a factor 1-∑(lx^2), and so the function is zero if the lx are direction sines and sum to 1. (Direction sines are used because the angles are measured towards the x direction and the lx contribution disappears when this angle is zero). Each additional space dimension adds the square of its direction sine. The PDE is therefore satisfied by any unit velocity function moving forward in any number of directions. This gives a Light-like limiting velocity in two possible ways:-
1. If pairs of directions fold to single dimensions, the disturbance travels at a speed that is less than unity unless there is a zero velocity in one of each of the folded pairs.
2. If all but 3 space directions are curled-up (Kaluza-Klein) directions, KK velocities reduce the velocity in the spatial directions; light-like disturbances have no velocity in the KK directions. KK "orbital" velocities give mass to the disturbances.

The spherically symmetric case is developed in http://www.mathpages.com/home/kmath242/kmath242.htm

(∂^2) (ψ/∂)(x^2) + (((n - 1)/r)) ∂(ψ/∂)r  =  (∂^2) (ψ/∂)(t^2)

5.3 KdVB Equation, Rational Pulse.

The non-linear conservative evolution KdVn (Korteweg-deVries-n) equation
 (Subscript[([(n^2) A ]),  t]) \(TraditionalForm\`\(+\(\([\ A\_\(x\ x\) + \(n(1 + n\ )\) \(A\^\(1 + 2/n\)\) A\_x]\)\(\ \)\)\_x\)\) = 0 i.e. (n^2)At+Axxx+(n+1)(n+2)(A^(2/n))Ax=0 (Zabulsky 1963 with  n->1/n ) combines KdV (n=2) and MKdV (n=1). It has well-known solitonic Sech solutions of the form u/uo = \(TraditionalForm\`\((\(±2\)\ j\ Sech\ \((\(±j\)\ \((\(-j\^2\)\ t/to + xi + x/xo)\))\))\)\^n\) (any n), and (((Jacobi Cnoid))^n) solutions for n=1 or 2. A search for rational and Gaussian pulse equations led to a broadened form conserving the additional property (Subscript[([((Subscript[A, x])^2)/A] ), x]) , and including  (Subscript[[(A^(1 + m))] , t]). Rational and Gaussian functions are solutions for the degenerate KdVn cases where n=-1 and n=-2 respectively. The final form proved to be
         \(TraditionalForm\`\([\ \(n(1 + n\ r)\) \(A\^\(1 + 2/n\)\) A\_x + \((r - 1)\) A\_x\^2/A + A\_\(x\ x\)]\)\_x + \([\(A\^\(1 + m\)\) \(\(n\^2\)(r + m)\)]\)\_\(\(\ \)\(t\)\)\  = 0\)   (KdVB)
Adding a y direction and the term (1 + n)((Subscript[y, 0])^2)/j  (Subscript[[(A^(1 - j))(Ay^j)], y]) , which differentiates to (1+n) ((Subscript[y, 0])^2)Ayy/(Subscript[A, 0]), gives the Kadometsev-Petviashvili (KP) equations when m=0, r=1, n=2. KP1 corresponds to (Subscript[y, 0])=i, KP2 to (Subscript[y, 0])=1. Note that (Subscript[A, xxx]) is related to the KP term, being  (Subscript[[(Subscript[[(A^(1 - j))(Ax^j)], x])/j], x]); both (Subscript[A, xxx]) and (Subscript[A, yy]) have infinite sets of conserved properties, characterised by j.

{x, y, A}

Off[General::"spell", 
      General::"spell1"]  ; Unprotect[x, j]; U =. ; j =. ; A =. ; X =. ; 
    Y =. ; {Simplify[
        D[U[t, y]^(1 - j)*D[U[t, y]^j, y]/j, 
          y]], 
Simplify[ 
        D[D[A[t, x]^(1 - j)*D[A[t, x]^j, x]/j, x], x]]}

Equation Synthesis for a TXY rational pulse is demonstrated:-
Define a pulse with peak amplitude a, breadth parameter b, exponent n, velocities kx, ky. Create a re-write rule to clarify expressions and aid simplification:-

v=(((a/(1+b(kx(((x + xi - lx t))^2)+ky(((y + yi - ly t))^2)))))^n);
\(vs\  = {kx \((x + xi - lx*t)\)^2\  + \ 
                    ky \((y + yi - ly*t)\)^2 -> \ kXY, 
\ 
                x + xi - lx*t -> \ X, \ lx*t - x - xi -> \ \(-X\), 
\ 
                y + yi - ly*t -> \ Y, \ ly*t - y - yi -> \ \(-Y\)};\)
V = Simplify[v /. vs]

Find derivatives and clarify:-

vx = D[v, x]; Vx = Simplify[vx /. vs];
vy = D[v, y]; Vy = Simplify[vy /. vs];
vt = D[v, t]; Vt = Simplify[vt /. vs];
vxx = D[v, {x, 2}]; Vxx = Simplify[vxx /. vs];
vyy = D[v, {y, 2}]; Vyy = Simplify[vyy /. vs];
vxxx = D[v, {x, 3}]; vyyy = D[v, {y, 3}];
VxxxT = Simplify[(vxxx + vyyy)/Vt /. vs];
Vxy3T = 
          Simplify[(2 (Vxx Vx + Vyy Vy) - (Vx^3 + Vy^3)/V)/
                V/Vt];

Substitute in the differential equation LHS with parameter p, giving an expression for the 2D rhs. kXY is reverted to the original, using the symbol Vo in place of V (necessary because V is an expression):-

rhs2d=Simplify[
            VxxxT + p Vxy3T] /. {kXY -> (a (Vo^(-1/n)) - 1)/b}

Essential simplifying relationships can be identified and applied, to give a simple RHS :-

Solve[1 + n + n p == 0, p]

Simplify[rhs2d /. {p -> -1 - 1/n}]

Simplifications are lx=kx, ly=ky,b=a/2:-

Simplify[
      rhs2d /. {p -> -1 - 1/n, lx -> kx, ly -> ky, 
          b -> a/2}]

Transfer this to the LHS and change the sign, to give the required differential equation
(1+n)Vt (V^(1/n)) - Vxxx - Vyyy +(1+1/n)(2(Vxx Vx+Vyy Vy) - (((Vx^3)+(Vy^3))/V)/V) = 0

Confirm correctness by using the conditions to re-define V, calculating the derivatives, and then showing that the equation simplifies to zero:-

V = 
        a/(1 + (a*(kx*(x + xi - kx*t)^2 + 
                        ky*(y + yi - ky*t)^2))/2); v = V^n;
vx = D[v, x]; vy = D[v, y]; vt = D[v, t]; vxx = D[v, {x, 2}]; 
      vyy = D[v, {y, 2}];
vxxx = D[v, {x, 3}];vyyy = D[v, {y, 3}];Simplify[(1 + n)vt V - vxxx - vyyy +(1 + 
                    1/n) (2 (vxx vx + vyy vy)  - (vx^3 + vy^3)/
                        v)/v==0]

The TXYZ case gives a related equation with summation over x,y,z ; the terms for each new dimension cancel. Putting ky and/or kz = 0 reduces to a lower dimensional case. The result is valid for any number, d, of space-like dimensions. I have failed to find either a stable integration or a modulated version.

Hence  (1+1/n)Vt (V^(1/n))+(∑_(x, y,  .. ))[-Vxxx+(1+1/n)(2Vx Vxx/V-Vx³/V²)] = 0 
has a solution      V=(((a/(1+a (∑_(x, y,  .. ))[kx (X+xi-kx T)²]/2/n )))^n)

5.4 Modulated Gaussian Pulse - KdVG Equation.

A similar procedure shows that:-

2 Gt + (∑_(x, y,  .. ))[ -Gxxx + 2Gx Gxx/G - Gx³/G²] = 0
or                                   [2G]t +(∑_(x, y,  .. ))[ [-Gxx + Gx²/G]x] = 0
has a solution    G = Exp[go+(∑_(x, y,  .. ))[i(ix (X-kxT)) - kx(X+xi-kx T)²]]

This is a stable travelling gaussian envelope with velocity components (jx^2), spread (jx^(-2)), modulation wave numbers kx, and angular velocity (∑_(x, y,  .. ))[kx (jx^2)]. It becomes an unmodulated pulse if the wave numbers are all zero. The result is valid for any number, "m" of space-like directions. The pulse has 4 balanced phases, Cos, Sin, -Cos, -Sin; plotting just the real and imaginary phases gives an unbalanced set of phases - the result of "folding" pairs of directions. Replacing i by 𝕤, with (𝕤^r)=1, gives modulation in r phases. 
	The amplitude depends on the free variable go. Attempts to raise the baseline did not lead to any useful simplifications (Some (Sech^2) pulses have velocities reduced by raised baselines.). The TXYZ relationship is confirmed; similar results are obtained for any number of spatial directions:-

G = 
      Exp[go + I (ix (x - jx t)) - 
          kx (((x + xi - lx t))^2) + I (iy (y - 
                    jy t)) - 
          ky (((y + yi - ly t))^2) + I (iz (z - 
                    jz t)) - kz (((z + zi - jz t))^2)]; Gt = D[G, t]; 
    Gx = D[G, x]; Gxx = D[Gx, x]; Gxxx = D[Gxx, x]; Gy = D[G, y]; 
    Gyy = D[Gy, y]; Gyyy = D[Gyy, y]; Gz = D[G, z]; Gzz = D[Gz, z]; 
    Gzzz = D[Gzz, z];
rhsxyz = 
      Simplify[{2  Gt - Gxxx + 2  Gx Gxx/G - (Gx^3)/(G^2) - Gyyy + 
            2  Gy Gyy/G - (Gy^3)/(G^2) - Gzzz + 2  Gz Gzz/G - 
            (Gz^3)/(G^2)}]

The imaginary component becomes zero if jx=kx etc; the real part becomes zero when lx=kx etc; ix, iy, & iz are unconstrained:-

Simplify[{rhsxyz, "G = ", G} /. { jx  -> kx, jy  -> ky, 
          jz -> kz, lx -> kx, ly -> ky, lz -> kz}]

5.5 Power Sinusoid, TXY.. travelling pulse.

The Power Sinusoid differential equation is (Overscript[v, ¨]) +(1/p - 1) ((\(v\& . \) )^2)/v+p (ω^2)v=0 and is solved by v=aSin[ωt + ϕ]^p; it reduces to the sinusoid equation (Overscript[v, ¨])+ (ω^2)v=0 when p=1. It generalises to the travelling wave power sinusoid, V = \(\((a\ Sin[t\ ω - \(Σ\_\(x, y,  .. \)\) kx\ \((x + xi)\)])\)\^p\) in any number of space directions XY.. as a solution of Vtt+(1/p - 1) Vt^2/V+p(ω^2)V. This is an expanding ripple of constant height. volume is not conserved because the DE is not a conservative form.

V=a ((Sin[ϕ + t ω - kx*(x + xi) - 
                      ky*(y + yi) - kz*(z + zi)])^p);Simplify[
        D[V, {t, 2}] + (1/p - 1) D[V, t]^2/V + 
          p (ω^2) V]

The p→∞ limit is the Gaussian pulse. The travelling wave power sinusoid is also a solution to the (conservative) KdVG equation, as shown below for the TXYZ case, but the simplifying constraints are {lx → (kx^2), ly → (ky^2), 
        lz → (kz^2)} etc.:-

V = 
        a Sin[ϕ + kx*(x + xi - jx*t) + ky*(y + yi - jy*t) + 
              kz*(z + zi - jz*t)]; v = V^p;
vx = D[v, x];vy = D[v, y];vz = D[v, z];vt = D[v, t];vxx = D[v, {x, 2}];vyy = D[v, {y, 2}];vzz = D[v, {z, 2}];
vxxx = D[v, {x, 3}];vyyy = D[v, {y, 3}];vzzz = D[v, {z, 3}];rhs=Simplify[p vt - (vxxx  +  vyyy + vzzz) +(1 - 
                    1/p) (2 (vxx vx + vyy vy + 
                            vzz vz)  - (vx^3 + vy^3 + vz^3)/v)/v]

rhs /. {jx -> (kx^2), jy -> (ky^2), jz -> (kz^2)}

\(Hence\ \ v = 
      a\ Sin[ϕ + \(Σ\_\(x, y,  .. \)\) kx\ \((x + xi - kx\^2\ t)\)]\^p\ solves\)
  p (v^2) vt + (Subscript[Σ, x, 
                y,  .. ]) (-(v^2) vxxx + (1 - 
                      1/p) (2  v vxx vx - v (vx^3))) = 
        0
\(giving\ a\ plane\ wave\ train\ in\ any\ direction\ in\ any\ number\ of\ \(\(dimensions\)\(.\)\)\)