---
title: A Numerical Tower For Geometric Algebra
packages: ['racketWithRackCheck']
---

<!--
FIXME: Can we extend an n-dimensional GA or PGA to a CGA? Seems to require an
    extra h, but which to choose if we're already using those for our direction
    vectors? Seems naff to reserve h0 or something since GA pays without needing
    it
TODO: Reader macros
TODO: Use provide, etc. to replace number?, zero?, +, etc.
TODO: Applications: include subalgebras corresponding to existing things, e.g.
  complex numbers, quaternions, etc. (hence we gain their applications too)
TODO: Using `geometric` to implement (vanilla) GA, PGA, CGA, etc.
-->

<!-- Unicode for copy/pasting:
×
SUB: ₀₁₂₃₄₅₆₇₈₉
SUP: ⁰¹²³⁴⁵⁶⁷⁸⁹ⁿ
-->

```{pipe="sh > /dev/null"}
# Commands to use in our pipe attributes: both append code to the end of geo.rkt
# but one also shows it on the page and the other doesn't.
{
  echo '#!'"$(command -v bash)"
  echo 'tee -a geo.rkt'
  echo 'echo >> geo.rkt'
  echo 'echo >> geo.rkt'
} > show

{
  echo '#!'"$(command -v bash)"
  echo 'cat >> geo.rkt'
  echo 'echo >> geo.rkt'
  echo 'echo >> geo.rkt'
} > hide

chmod +x show hide
```

**Note:** In this post I will avoid my preferred [overbar
notation](/projects/units/negative_bar_notation.html), and instead write
negatives using a "minus sign" (like $-123$) for consistency with Scheme/Racket
notation.

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## Introduction ##

I've recently taken an interest in [geometric algebra](https://bivector.net)
("GA", not to be confused with [algebraic
geometry](https://en.wikipedia.org/wiki/Algebraic_geometry)!), which goes beyond
the usual ["number line"](https://en.wikipedia.org/wiki/Number_line) in a way
that elegantly models geometric ideas such as circles, volumes, rotations,
etc. I don't want to motivate or advocate for GA itself, since there are plenty
of [much better resources](https://bivector.net/doc.html#five) out there. Hence
I'm going to focus on explanation and implementation of some basic foundations
of GA, since that's a good way to check my own understanding of the subject!

 Here's the start of the implementation, to get us
going:

```{.scheme pipe="./show"}
#lang racket
```

```{pipe="./hide"}
;; We'll write our documentation using Scribble
(require scribble/srcdoc
         (for-doc racket/base scribble/manual))

;; We'll define a test suite as we go, as a "sub-module" called 'test'
(module+ test (require rackunit rackcheck-lib))
```





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