An alternative to vector/matrix algebra ideal for (amongst many other things) geometry.

An algebra on a graded linear (vector) space, with an invertible product that operates on multivectors.

We will first tackle the Algebra.

الكتاب المختصر في حساب الجبر والمقابلة

al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah

Muḥammad ibn Mūsā al-Khwārizmī

$= 39$

$+ 25=64$

$(x+5)^2 = 64$

$\Leftrightarrow x = 3$

$ax^2 = bx$

$ax^2 = c$

$bx = c$

$ax^2 + bx = c$

$ax^2+c = bx$

$bx + c = ax^2$

Why six equations ?????

No negative numbers, no zero !!

$x + 1 = 1$

$x + 1 = 0$

zero!

negative numbers!

With 0 and negative numbers, we reduce the amount of equations from 6 to 1 !!!

$x^2 \in \mathbb R$

$(\alpha \mathbf e_1 + \beta \mathbf e_2)^2 \in \mathbb R$

$\alpha^2 \mathbf {e_1}^2 + \alpha\beta\color{lightgreen}\mathbf e_1\mathbf e_2\color{white} + \alpha\beta\color{lightgreen}\mathbf e_2\mathbf e_1 \color{white} + \beta^2 {\mathbf e_2}^2 \in \mathbb R$

$\color{lightgreen} \mathbf e_1\mathbf e_2 + \mathbf e_2\mathbf e_1 \color{white} = 0$

$\mathbf e_1\mathbf e_2 = - \mathbf e_2\mathbf e_1 \color{white}$

$\alpha\mathbf e_1 + \beta\mathbf e_2$

$\alpha\mathbf e_{12}$

$\alpha\mathbf e_{12}$

$\alpha\mathbf e_{123}$

$v_1v_2 = (\color{lightblue}x_1\color{white}\color{lightgreen}x_2\color{white}+\color{lightblue}y_1\color{white}\color{lightgreen}y_2\color{white}+\color{lightblue}z_1\color{white}\color{lightgreen}z_2\color{white}) + (\color{lightblue}x_1\color{white}\color{lightgreen}y_2\color{white}-\color{lightblue}y_1\color{white}\color{lightgreen}x_2\color{white})\mathbf e_{12} + (-\color{lightblue}x_1\color{white}\color{lightgreen}z_2\color{white}+\color{lightblue}z_1\color{white}\color{lightgreen}x_2\color{white})\mathbf e_{31} + (\color{lightblue}y_1\color{white}\color{lightgreen}z_2\color{white}-\color{lightblue}z_1\color{white}\color{lightgreen}y_2\color{white})\mathbf e_{23}$

$\vec v_1 = \begin{bmatrix}\color{lightblue}x_1 \\ \color{lightblue} y_1 \\ \color{lightblue} z_1  \color{white} \end{bmatrix} \quad \vec v_2 = \begin{bmatrix}\color{lightgreen}x_2 \\ \color{lightgreen} y_2 \\ \color{lightgreen} z_2  \color{white} \end{bmatrix}$

$v_1v_2 = (\color{lightblue}x_1\color{white}\color{lightgreen}x_2\color{white}+\color{lightblue}y_1\color{white}\color{lightgreen}y_2\color{white}+\color{lightblue}z_1\color{white}\color{lightgreen}z_2\color{white}) + (\color{lightblue}x_1\color{white}\color{lightgreen}y_2\color{white}-\color{lightblue}y_1\color{white}\color{lightgreen}x_2\color{white})\mathbf e_{12} + (-\color{lightblue}x_1\color{white}\color{lightgreen}z_2\color{white}+\color{lightblue}z_1\color{white}\color{lightgreen}x_2\color{white})\mathbf e_{31} + (\color{lightblue}y_1\color{white}\color{lightgreen}z_2\color{white}-\color{lightblue}z_1\color{white}\color{lightgreen}y_2\color{white})\mathbf e_{23}$

Note : One scalar and three imaginary parts. The geometric product of two vectors in $\mathbb R_3$ is a quaternion!

The highest and lowest grade parts are often useful and are called the wedge and dot products respectively.

$\langle a \rangle_s \cdot \langle b \rangle_t = \langle ab \rangle_{|s-t|}$

$\langle a \rangle_s \wedge \langle b \rangle_t = \langle ab \rangle_{s+t}$

$\begin{aligned} \gamma_0 &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \ \gamma_1 &= \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{bmatrix} \ \\ \gamma_2 &= \begin{bmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{bmatrix} \ \gamma_3 &= \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \end{aligned}$

$\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \ \sigma_2 = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \ \sigma_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

$\gamma_0 = \mathbf e_0\,\, \gamma_1 = \mathbf e_1\,\, \gamma_2 = \mathbf e_2\,\, \gamma_3 = \mathbf e_3$

$\sigma_i = \gamma_i \gamma_0 \implies \newline \sigma_1 = \mathbf e_{10},\,\,\,\sigma_2 = \mathbf e_{20},\,\,\,\sigma_3 = \mathbf e_{30}$

$I = \gamma_0 \gamma_1 \gamma_2 \gamma_3 = \sigma_1 \sigma_2 \sigma_3$

$\begin{aligned} \gamma_0 &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \to \mathbf{e}_1, \quad \gamma_1 &= \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{bmatrix} \to \mathbf{e}_2 \\ \gamma_2 &= \begin{bmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{bmatrix} \to \mathbf{e}_3, \quad \gamma_3 &= \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \to \mathbf{e}_4\end{aligned}$

$\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad \sigma_2 = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \quad \sigma_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

$\sigma_i = \gamma_i \gamma_0$

The definitive example of the power of the spacetimeis the singular Maxwell equation:

$$\partial F = J \quad \implies \quad \partial \cdot F = J \quad \partial \wedge F = 0$$

where $\partial = \gamma^\mu \partial_\mu$ is the gradient operator, $J = \gamma^\mu J_\mu$ is the source term, and $F = \mathbf{E} + I \mathbf{B}$ is the field strength bivector. To prove that this indeed reproduces the familiar Maxwell equations, we use the spacetime split:

$$\begin{aligned} J \gamma_0 &= J \cdot \gamma_0 + J \wedge \gamma_0 = \rho + \mathbf{J} \\ \partial \gamma_0 &= \partial \cdot \gamma_0 + \partial \wedge \gamma_0 = \partial_t + \nabla \end{aligned}$$

Most compact form of the Maxwell equations!

$\partial \cdot F = J \leftrightarrow \partial_\alpha F^{\alpha \beta} = J^\beta$

$\partial \wedge F = 0 \leftrightarrow \partial_\alpha (\tfrac{1}{2} \epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}) = 0$

$\mathbb R_{3,0,1}$

$\mathbb R_{4}$

$\mathbb R_{3,1}$

FLAT

SPHERICAL

HYPERBOLIC

$ab$

compose any a,b

$a\wedge b$

intersect any a,b

$a\vee b$

join any a,b

$(a\cdot b)b^{-1}$

project any a onto/into b

for any point/line/plane a,b

A very natural language to describe the geometry of points, lines and planes.

(can be bent into the conformal group and the geometry of spheres, circles and point-pairs)

$\langle a \rangle_s \cdot \langle b \rangle_t = \langle ab \rangle_{|s-t|}$

$\langle a \rangle_s \wedge \langle b \rangle_t = \langle ab \rangle_{s+t}$

$( a \vee b)^* = a^* \wedge b^*$

$a \times b = \frac 1 2 (ab - ba)$

Wedge

Dot

Vee

Commutator