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

A floretion is a type of hypercomplex number that extends the concept of quaternions. One main characteristic of floretions is their representation in an octal number system, where the digits are limited to {1, 2, 4, 7} which, when endowed with multiplication, become a discrete mathematical group. The number of digits in octal is the "order" of the group. These numbers can be associated with geometric shapes, specifically equilateral triangles, which are used to represent complex operations and properties visually.

In mathematical operations, floretions employ logic operations like XOR for multiplication, following rules akin to quaternion multiplication but extended into higher dimensions and complexity. Floretions can be manipulated algebraically or graphically, each method offering unique insights into their structure and behavior.

The development and use of floretions incorporate both algebraic structures and geometric representations, bridging a unique connection between number theory and visual geometry.

Floretion Calculator: floretions.com

Floretions at GitHub

Dataset Description for Floretion Multiplication (up to order 8):

This collection of datasets, stored in numpy binary format, is designed to facilitate efficient multiplication of floretions up to order 8. The matrices included—indices and signs—are optimized for rapid computation of the product z=x⋅y using standard dot products, enhancing both speed and efficiency.

For a foundational understanding, consider quaternion multiplication (order 1 floretions):

x = x[0]i + x[1]j + x[2]k + x[3]e 

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

Multiplication Outcome: The coefficient for base vector "i" in z=x⋅y is calculated as:

x[0]y[3]+x[1]y[2]−x[2]y[1]+x[3]y[0]=np.dot(np.multiply([1,−1,1,1],[x[3],x[2],x[1],x[0]]),[y[0],y[1],y[2],y[3]])

Notice that [1,−1,1,1] above is the first row in the "signs" matrix and the indices of the coefficients of x are given by the first row of the "indices" matrix, below. 

"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 streamline the computation of floretion products across various orders. For order 7 floretions, the data is split into 4 segments, for order 8 into 64 segments to keep file sizes managable.