Introduction

Procedurally generated animation of the anti-twister mechanism and its connection to Spin(3). Also known as Dirac's belt trick, a demonstration of an object that is subject to $4\pi$ or $720°$ symmetry, so it needs two full revolutions to revert to its initial state.

Inspired by Jason Hise's animations, please check out: https://en.wikipedia.org/wiki/User:JasonHise

States of the anti-twister and their corresponding spin observables as quaternions

How it works

Coded using CGA ($\mathrm{Cl(4,1)}$) motor interpolation as described by Belon et al (2017).

In order to model the ribbon that is secured in $\mathbf{s}$-direction while rotating in $\mathbf{r}$ by $2\pi\cdot\lambda \mathrm{rad}$. We define three oriented control points using CGA rotors. For this, using two rotors $R$ and $S$, describing the rotation of the center cube and the twisting of the ribbon $$\large R(\lambda) = \exp(\frac{\mathbf{r}}{e_{123}} \pi \lambda),$$ $$\large S = \exp(-\frac{\mathbf{s}}{e_{123}} \frac{\pi}{2})$$ and three translators $$\large T_0(\lambda) = 1- \frac{R(\lambda) 0.3 \mathbf{s} R(\lambda)^\dagger \wedge e_\infty}{2},$$ $$\large T_1(\lambda) = 1- \frac{R(\lambda) 1.0 \mathbf{s} R(\lambda)^\dagger \wedge e_\infty}{2},$$ $$\large T_2 = 1- \frac{2.0 \mathbf{s} \wedge e_\infty}{2} $$ we define the motors of oriented control points as $$\large M_0(\lambda) = T_0(\lambda) R(\lambda) S $$ $$\large M_1(\lambda) = T_1(\lambda) R(\lambda) S $$ $$\large M_2(\lambda) = T_2 $$ Which can be interpolated linearly using motor logarithms: $$\large M(\lambda,\alpha) = \exp(\sum_i B_i(\alpha) \log(M_i(\lambda)))$$ Where $\alpha \in \left[0,1\right]$ is the interpolation parameter along the ribbon originating from the center cube face. And $B_i(\alpha)$ are weight functions defined using $$\large B_i(\alpha) = B_i'(\alpha) / \sum_j B_j'(\alpha) $$ which is normalizing $$\large B_1'(\alpha) = (1-\alpha)^2,$$ $$\large B_2'(\alpha) = 10 \alpha^2 (1-\alpha)^4,$$ $$\large B_3'(\alpha) = \alpha^3$$.

Finally, the interpolation motor $M(\lambda,\alpha)$ can be used to calculate the mesh of a ribbon extending in $\mathbf{c}$ direction $$\large \rho_{l,r}(\lambda,\alpha) = M(\lambda,\alpha) (\pm \uparrow \mathbf{c}) M(\lambda,\alpha)^\dagger.$$ Where $\rho_{l}(\lambda,\alpha)$ and $\rho_{r}(\lambda,\alpha)$ are its left and right boundaries respectively and we used the up-projection $\uparrow \mathbf{c}$ for the conformal representation defined by $\uparrow \mathbf{c} = \mathbf{c} + \frac{1}{2}\mathbf{c}^2 e_\infty + e_o $.

The full set of twelve equations (2 boundaries $\times$ 6 directions) for rotating in the $z$-axis is given by

$$\large\rho^{+ x}_{l,r}(\lambda,\alpha) = \underline{M(\lambda, \alpha,\mathbf{s}= +e_1,\mathbf{r}=e_3)} (\uparrow \pm 0.1e_2)$$ $$\large\rho^{- x}_{l,r}(\lambda,\alpha) = \underline{M(\lambda + 1,\alpha,\mathbf{s}=-e_1,\mathbf{r}=e_3)} (\uparrow \pm 0.1e_2)$$ $$\large\rho^{+ y}_{l,r}(\lambda,\alpha) = \underline{M(\lambda + \frac{3}{2} ,\alpha,\mathbf{s}=+e_2,\mathbf{r}=e_3)} (\uparrow \pm 0.1e_3)$$ $$\large\rho^{- y}_{l,r}(\lambda,\alpha) = \underline{M(\lambda + \frac{1}{2} ,\alpha,\mathbf{s}=-e_2,\mathbf{r}=e_3)} (\uparrow\pm 0.1e_3)$$ $$\large\rho^{+ z}_{l,r}(\lambda,\alpha) = \underline{R_{12}(\frac{\lambda}{2})M(\frac{1}{2},\alpha,\mathbf{s}=+ e_3,\mathbf{r}=e_2)) R_{12}(-\frac{\lambda}{2})} (\uparrow\pm 0.1e_1)$$ $$\large\rho^{- z}_{l,r}(\lambda,\alpha) = \underline{R_{12}(\frac{\lambda}{2})M(\frac{3}{2},\alpha,\mathbf{s}=- e_3,\mathbf{r}=e_2)) R_{12}(-\frac{\lambda}{2}) } (\uparrow\pm 0.1e_1).$$

Where we used $\underline{M}(x)$ to denote the sandwich-product $MxM^\dagger$ of $x$. The $\large\rho^{\pm i}_{l,r}(\lambda,\alpha)$ are at last projected back into $\mathrm{Cl(3,0)}$ by using

$$\downarrow\large \rho = (\frac{\rho}{\rho \cdot e_\infty} \wedge e_+ \wedge e_-)(e_+ \wedge e_-)^{-1}$$

Which first normalizes the conformal point by dividing it with $\rho \cdot e_\infty$ and then rejects it from the Minkowski plane $e_+ \wedge e_-$.

How to it compile yourself

After cd in the project folder download dependencies using

pip install -r requirements.txt

After that, you may use pyinstaller to build an executable

pyinstaller ./Spinor_Cube_Ver2.2.2.py --onefile

References:

[1] Belon, M.C.L., Hildenbrand, D. Practical Geometric Modeling Using Geometric Algebra Motors. Adv. Appl. Clifford Algebras 27, 2019–2033 (2017). https://doi.org/10.1007/s00006-017-0777-z