hurwitz 1.1.1

A module for working with Hurwitz quaternions

<img src="https://static.wixstatic.com/media/355b75_1c4e29d87f1e449cbdfdb2b623ac66ce~mv2.png/v1/fill/w_292,h_72,fp_0.50_0.50,q_85,usm_0.66_1.00_0.01,enc_auto/355b75_1c4e29d87f1e449cbdfdb2b623ac66ce~mv2.png" width="100"> research, Inc. - Hurwitz

Hurwitz

A Hurwitz Quaternion Integer Library for Python - Made by Harper Chisari

Installation

Currently, Hurwitz is easy to install, as it has no dependencies. This is likely to change soon with a significant extension on the way.

You can install the Hurwitz package from PyPI using pip:

pip install hurwitz

Usage

First, import the HurwitzQuaternion class from the Hurwitz package:

from hurwitz import HurwitzQuaternion

You can create a Hurwitz quaternion by instantiating the HurwitzQuaternion class with four integer values:

q1 = HurwitzQuaternion(1, 2, 3, 4) # 1 + 2i + 3j + 4k
q2 = HurwitzQuaternion(2, 3, 4, 5) # 2 + 3i + 4j + 5k

If you want to create a half quaternion, set the half parameter to True:

q_half = HurwitzQuaternion(1, 3, 5, 7, half=True)

[!IMPORTANT] Half-Integer quaternions cannot accept zeroes nor even values as coefficients. Those would lead to a non-Hurwitz quaternion. If you want to get to the 'closest' quaternion, use the snap method below.

Basic Operations

Simple operations on Hurwitz Integers are easy in Hurwitz!

Addition, Subtraction, and Multiplication

q3 = q1 + q2
print(q3)  # Output: 3 + 5i + 7j + 9ij

q4 = q1 - q2
print(q4)  # Output: -1 + -1i + -1j + -1ij

q5 = q1 * q2
print(q5)  # Output: -36 + 6i + 12j + 12ij

q6 = q1 * 2
print(q6)  # Output: 2 + 4i + 6j + 8ij

Hurwitz currently only supports integer multiplication, mostly because it made subtraction easier to implement.

Norm, Conjugate, and Inverse

Norm

The norm in Hurwitz is defined as:

$$ \text{Nr}(q) = \begin{cases} \sqrt{a^2 + b^2 + c^2 + d^2} & \text{if } q \text{ is a whole quaternion} \ \sqrt{\left(\frac{a}{2}\right)^2 + \left(\frac{b}{2}\right)^2 + \left(\frac{c}{2}\right)^2 + \left(\frac{d}{2}\right)^2} & \text{if } q \text{ is a half quaternion} \end{cases} $$

and can be simply calculated with

q1.norm()

Conjugate

The conjugate of a Hurwitz quaternion ( q = a + bi + cj + dk ) is defined as:

$$ \text{Conj}(q) = a - bi - cj - dk $$

For half quaternions, the conjugate is:

$$ \text{Conj}(q) = \frac{a}{2} - \frac{b}{2}i - \frac{c}{2}j - \frac{d}{2}k $$

You can calculate the conjugate using q1.conjugate()

Inverse

The inverse of a unitary Hurwitz quaternion ( q ) is defined as the conjugate of ( q ) divided by the norm of ( q ) squared.

For Hurwitz quaternions, this includes only the 24 units:

$$ \pm 1, \pm i, \pm j, \pm k, \frac{ \pm 1 \pm i \pm j \pm k}{2} . $$

For a non-unitary quaternion, the general inverse is calculated as:

$$ \text{Inv}(q) = \frac{\text{Conj}(q)}{\text{Nr}(q)^2} $$

For unitary quaternions, you can calculate the inverse using q1.inverse()

Example Code

Here’s how you can demonstrate these calculations using the hurwitz package:

from hurwitz import HurwitzQuaternion

# Create a Hurwitz quaternion
q1 = HurwitzQuaternion(1, 2, 3, 4)

# Calculate the norm
norm_q1 = q1.norm()
print("Norm of q1:", norm_q1)

# Calculate the conjugate
conj_q1 = q1.conjugate()
print("Conjugate of q1:", conj_q1)

# Calculate the inverse (only if q1 is unitary)
try:
    inv_q1 = q1.inverse()
    print("Inverse of q1:", inv_q1)
except ValueError as e:
    print(e)

General Quaternion Operations

Hurwitz does support transformation to 'general' quaternions in the form of a 4-tuple. In order to do so, simply invoke the general method:

q1_general = q1.general
print(q1_general) # prints a tuple (1, 2, 3, 4)

This includes the general_inverse, general_quaternion_multiplication, and general_quaternion_division methods. These naturally also return 4-tuples.

Advanced Operations

Division

Yes! Division of Hurwitz integers! Admittedly, just as with regular integers, this is implemented as Euclidean division.

The Euclidean division can either be accessed directly as euclidean_division or in components using the // floordiv and % modulo operators.

q7 = q1 // q2  # Floor division
print(q7)  # Output: 2

q8 = q1 % q2  # Modulus
print(q8) # Output: -2 - 2i - 2j - 2k

q9 = q1.euclidean_division(q2)  # Euclidean division
print(q9)  # Output: (2, -2 - 2i - 2j - 2k)

q8 = q1 / q2  # True division
print(q8)  # Output: (0.74, 0.037, 0.0, 0.074)

Citation: For more on Euclidean division in the context of Hurwitz quaternions and to see the source of the algorithm used, see: Boyd Coan and Cherng-tiao Perng, "Factorization of Hurwitz Quaternions," International Mathematical Forum, Vol. 7, 2012, no. 43, 2143 - 2156.

(Int) Powers

Currently, you can raise Hurwitz quaternions to integer powers. In the future we will add quaternion power support, and potentially fractional and logarithm support.

Decomposition

Decomposition is simply a method for decomposing a Hurwitz quaternion into unitary ones. This generally consists of a few whole Hurwitz quaternions and a single half quaternion unit.

print("Decomposition of q1:", q1.decompose()) # results in 1 + 2 + 3 + 4 = 10 whole unit Hurwitz quaternions

q_half_ex = HurwitzQuaternion(1,3,5,7, half=True) # 1/2 + 3/2 i + 5/2 j + 7/2 k

print("Half quaternion:", repr(q_half_ex))
print("Decomposition of q1:" )
for val in q_half_ex.decompose():
    print(repr(val)) 
# Output: 0 + 1i + 0j + 0ij, 0 + 0i + 1j + 0ij, 0 + 0i + 1j + 0ij0 + 0i + 0j + 1ij
# 0 + 0i + 0j + 1ij, 0 + 0i + 0j + 1ij, 1/2 + 1/2i + 1/2j + 1/2ij

Associates and Equivalence Classes

Associates of a Hurwitz Quaternion are defined as the quaternion multiplied by any of teh unit quaternuions, meaning that the Norm stays the same.

You can find all associates of a quaternion:

associates = q1.associates()
print("Associates of q1:", associates)

You can also find the equivalence class, the associates of a quaternion and it's conjugate:

equivalence_class = q1.equivalence_class()
print("Equivalence Class of q1:", equivalence_class)

Association Check

To check if two quaternions are associates:

q3 = HurwitzQuaternion(2, 3, 4, 5)
is_associate = q1.association_check(q1, q3)
print("Are q1 and q3 associates?", is_associate)

Bonus Features

Binomial Decomposition

You can can the binomial_composition() method to retrieve a tuple of the largest whole hurwitz quaternion and smallest half integer quaternion which can be added to get the given quaternion.

Euclidean Division Pro Max

This method attempts to divide the conjugate of the dividend by the divisor and returns the result with the smallest remainder:

q7_pro, r_pro = q1.euclidean_division_pro_max(q2)
print("Quotient (Pro Max) (q7_pro):", q7_pro)
print("Remainder (Pro Max) (r_pro):", r_pro)

Snap: Rounding to the Nearest Hurwitz Quaternion

The snap method rounds the values of a general quaternion to the nearest integer or half:

general_q = (1.2, 2.8, 3.5, 4.1)
q_snapped = q1.snap(general_q)
print("Snapped Quaternion:", q_snapped)

Symbolic String

The Hurwitz library also supports a new symbolic string representation via the symbolic_rep method. This is for something down the road!

Planned Features

In the near-future the library will support:

• H. Quaternion matrices and operations
• H. Quaternion factorization and Norm rings
• more primality-related features
• C-based, and CUDA performance improvements for parallel operations
• Qiskit integration for hypercomplex-quantum mechanics
• Generalized operations on Clifford Algebras

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