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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()}")