Indices and sign arrays for floretion multiplication up to order 8.

A floretion of order n is a real linear combination of signed basis words of length n in the alphabet {1, 2, 4, 7}. The four digits are identified with the quaternionic basis symbols

1 ↔ i, 2 ↔ j, 4 ↔ k, 7 ↔ e,

and multiplication is performed digit by digit, collecting the local signs into one global sign. In order one this recovers the quaternion group Q8. In general, the signed basis group is the central product of n copies of Q8; the value of the floretion representation is its explicit digitwise, computational, and geometric coordinate system.

The same alphabet {1, 2, 4, 7} also has a geometric interpretation. The digits 1, 2, and 4 label the three corner subtriangles in a recursive equilateral-triangle subdivision, while the digit 7 labels the central subtriangle. This gives a natural connection between the algebraic basis words and triangular tilings.

The multiplication rule admits a Boolean formulation using elementary bitwise operations such as XNOR and AND. This is not a new multiplication replacing quaternion multiplication; rather, it is the quaternion table written in Boolean coordinates. This representation is useful computationally because it allows multiplication in higher orders to be implemented by precomputed index and sign arrays.

Floretion Calculator 

Floretions at GitHub

Current draft paper

Dataset Description for Floretion Multiplication, up to order 8

This dataset contains precomputed index arrays and sign arrays, stored in NumPy binary format, for efficient multiplication of floretions up to order 8.

For two floretions x and y, the product z = x*y can be computed coefficient by coefficient. For each output basis vector z_r, the corresponding coefficient is obtained as an ordinary dot product between the coefficient vector of y and a signed, reordered version of the coefficient vector of x. The noncommutativity of floretion multiplication is encoded in the row-dependent index and sign arrays.

In other words, each row of the indices array specifies how the coefficients of x should be reordered for a given output basis vector, and the corresponding row of the signs array specifies which signs must be applied before taking the dot product with y.

For a simple order-one example, write

x = x[0]i + x[1]j + x[2]k + x[3]ey = y[0]i + y[1]j + y[2]k + y[3]e.

The coefficient of i in z = x*y is

x[0]y[3] + x[1]y[2] - x[2]y[1] + x[3]y[0].

This can be computed as

np.dot(np.multiply([1, -1, 1, 1], [x[3], x[2], x[1], x[0]]), [y[0], y[1], y[2], y[3]]).

Here [1, -1, 1, 1] is the first row of the signs matrix, while [3, 2, 1, 0] is the first row of the indices matrix.

For order one, the matrices are:

Signs matrix

[[ 1 -1  1  1] [ 1  1 -1  1] [-1  1  1  1] [-1 -1 -1  1]]

Indices matrix

[[3 2 1 0] [2 3 0 1] [1 0 3 2] [0 1 2 3]]

These precomputed matrices make it possible to replace repeated symbolic basis multiplication by array indexing, sign changes, and standard dot products. For larger orders, this becomes especially useful because the number of basis vectors grows as 4^n. To keep file sizes manageable, the order 7 data is split into 4 segments, and the order 8 data is split into 64 segments.