Napoleonic Vectors
Napoleon's Theorem states that the centers of three equilateral triangles constructed on the edges of any given triangle form an equilateral triangle. In the note on Napoleon's Theorem we saw that this proposition can be expressed in terms of the three complex numbers v1, v2, v3 representing the vertices of the given triangle in the complex plane. In general, three complex numbers z1, z2, z3 are the vertices of an equilateral triangle if and only if
From this, given any two vertices of an equilateral triangle, we can solve for the third, choosing the appropriate root, depending on whether we want a clockwise loop or a counter-clockwise loop. The centers of the counter-clockwise equilateral triangles are then given by the averages of their vertices, so the centers are given by
The differences between these centers are
Essentially Napoleon's Theorem asserts that the sum of the squares of these three quantities vanishes for any values of v1, v2, v3, and this is easily verified algebraically. Notice that the coefficients of the vertices are simply the cube roots of 1. Denoting these roots by r1, r2, r3, we have the identities
Hence the sum of squares of the three preceding differences is
To see why this sum vanishes, note the general algebraic identity
(1)
If we set x,y,z equal to the cube roots of 1, then x2 + y2 + z2 and xy + xz + yz both vanish, so the sum vanishes, regardless of the values of a,b,c.
This suggests an interesting algebraic structure. Let B denote the set of all ordered triples (x,y,z) of complex numbers such that xy + xz + yz = 0. An equivalent condition, for non-zero components, is 1/x + 1/y + 1/z = 0. Obviously (r1, r2, r3) is an element of B, but there are infinitely many others, such as (-2,3,6) and (-1,2,2). In general we have the two-parameter family [x, y, -xy/(x+y)]. We define the "norm" of an element X = (x1,x2,x3) of B as
Because of the trinary symmetry of the components, the norm can also be expressed in the equivalent form
Also, using the relation x3 = -x1x2/(x1+x2) we can eliminate x3 and write the norm as (x12 + x1x2 + x22)/(x1 + x2) and similarly for the other two pairs of indices. One consequence of this reduction is that if the components of X are integers (rational), then the norm of X is an integer (rational).
We can also define the "upward product" of two elements X and Y as
Denoting this product by Z = (z1, z2, z3), we can evaluate the sum
Since X and Y are both elements of B, it follows that this quantity vanishes, and so the upward product of X and Y is also an element of B. Furthermore, again taking advantage of the fact that X and Y are elements of B, it's clear from identity (1) that the norm is multiplicative, i.e., we have
Likewise the "downward product" defined by
is also an element of B, and the norm is multiplicative under this operation. Naturally the corresponding "division" operations can be performed by solving the three linear equations in three unknowns. For example, if (a,b,c)(x,y,z) = (A,B,C), and we are given the values of x,y,z,A,B,C, we can solve for a,b,c using the relation
The determinant of the square matrix is x3 + y3 + z3 - 3xyz, which for Napoleonic triples is equivalent to (x + y + z)3. Hence division is well-defined as long as x + y + z is not zero, and we can solve the above system to give
and similarly for b and c (by rotating the coefficients A,B,C to the right). Of course, the vanishing of the denominator corresponds to the vanishing of the norm of (x,y,z), which is x2 + y2 + z2. In view of the identity x2 + y2 + z2 = (x + y + z)2 - 2(xy+xz+yz) we see the norm of a Napoleonic triple vanishes if and only if the sum of its components is zero.
To define an analog of addition for elements of B is slightly less straightforward, because a simple addition of corresponding components of two elements of B generally does not yield an element of B. In other words, if X and Y are elements of B, the triple of numbers x1 + y1, x2 + y2, x3 + y3 is generally not in B. One way of defining "addition" for elements of B is by the formula
where the scalar l is given by
If Y is the zero element of B, i.e., Y = (0,0,0), then l = 0 and we have
On the other hand, if the two summands are identical, the numerator of l vanishes, so we have
with the understanding that multiplication of a Napoleonic vector by a scalar simply multiplies each component by that scalar.
The upward product is obviously not commutative, because the second and third components are transposed. In other words, if XY = (z1, z2, z3) then XY = (z1, z3, z2). The commutator can then be found by applying Napoleonic addition to XY and -(XY). The value of l for this operation is
(In the last step we have made use of the fact that 2z1z2 + 2z1z3 equals -2z2z3.) Inserting this into the addition formula gives the commutator
Interestingly, the norm of this vector is zero, since it's just a multiple of the fundamental "equilateral" vector consisting of the cube roots of 1. It's interesting that this is formally very similar to the commutator operator in quantum mechanics. If X,Y denote the operators corresponding to two incompatible observables (such as position and momentum), then a measurement on the state j of X followed by a measurement of Y is denoted by XY, whereas a measurement of Y followed by a measurement of X is denoted by YX, and these two compound operations do not commute. Specifically, we have
In a sense, the fundamental vector r = (r1, r2, r3)/
and its permutations play the role of Planck's constant 
= h/2p, i.e., they represent the minimal Napoleonic vector, having a norm of zero. The basic commutation relation of Napoleonic vectors can be written in the form
We also have
which we note is simply the signed area of the triangle whose vertices are (x1,y1), (x2,y2), and (x3,y3). (See the note on Net Area and Green's Theorem.) Of course, in the present case the components of X and Y are allowed to be complex, but we can still formally regard a pair of complex numbers as defining a point in the C2 "plane", and define the "area" accordingly. Denoting this area by Axy, we have
This shows that the "area" of the complex triangle defined by two given vectors plays a role analogous to that of the state vector (or wave function) in the quantum dynamical commutator relations. To carry this through, we could define X as an operator which, when applied to an "area" such as Axy gives the Napoleonic vector (x1,x2,x3), and similarly for Y. These operators, together with the definitions of multiplication and addition for Napoleonic vectors, would then enable us to write
