Floretions: A Playground for Algebra and Geometry

Introduction

This repository serves as a playground for exploring the algebraic structure of Floretions—a unique number system with roots in base-8 digits 1,2,4,7. Floretion Calculator.

Content Overview

• floretion.py: Implements the Floretion class to perform fundamental operations like creation and multiplication.
• SierpinskiFlo.py: Pass this class an object of type Floretion to be represented as a triangle tiling.
• data/: Directory containing CSV files to directly import base vectors of order 9 and less (as opposed to calculating these "on the fly", which could take a lot of time).

Classes

Class: Floretion

Represents a Floretion, a type of hypercomplex number that extends the concept of quaternions.

Attributes

• base_to_nonzero_coeff: Maps base vectors (in decimal) to their non-zero coefficients.
• format_type: String representing the format ('dec' or 'oct') of base vectors.
• max_order: Maximum order for the floretions.
• flo_order: The floretion order.
• grid_flo_loaded_data: Pandas DataFrame holding loaded data.
• base_vec_dec_all: NumPy array of all base vectors in decimal.
• coeff_vec_all: NumPy array of coefficients aligned with base_vec_dec_all.
• base_to_grid_index: Maps base vectors to their grid indices.

Methods

• __init__: Initializes a Floretion instance.
• from_preloaded_data, from_cartesian_coords: Methods for creating Floretions from different data types.
• find_flo_order: Calculates the floretion order.
• Arithmetic operations: __pow__, __add__, __sub__, __eq__, __mul__, __rmul__.
• mult_flo_base_absolute_value, mult_flo_sign_only: Methods for multiplying floretion base vectors.
• compute_possible_vecs: Computes possible vectors.
• as_floretion_notation: Converts to floretion notation.
• flo_oct_to_grid, grid_to_flo_oct: Conversion between floretion octal representation and grid format.
• normalize_coeffs: Normalizes coefficients.
• sum_of_squares, abs: Calculates sum of squares and absolute value.
• from_string: Creates a Floretion from a string representation.
• display_as_grid: Displays the Floretion as a grid.
• grid_to_coordinates: Converts grid format to coordinates.
• find_center: Finds the center of a Floretion.

Example Usage

flo_x = Floretion.from_string(".5iii +.5jjj + .5kkk")
flo_y = Floretion.from_string("iij + iik + jji +jjk + kki + kkj")

flo_z = flo_x * flo_y - flo_y * flo_x
print(f"flo_z {flo_z.as_floretion_notation()}")