HEISENBERG RENDERS

Below are some visualizations for the continuous Heisenberg group $H(\mathbb{R})$ that I used in my dissertation work.

In all cases the Heisenberg group is being viewed under exponenti al coordinates, i.e. as $\mathbb{R}^3$ with the group law

$(x_1, y_1, z_1) * (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2 + (1/2)(x_1y_2 - x_2y_1) )$.

Stl files are used for 3d printing. I myself have done so on a Dimension 1200es with some help from the Bray Lab at Tufts.

The metrics under consideration are Carnot Carathéodory metrics defined by norms on the horizontal distribution.

The sub-Riemannian metric is defined by the Euclidean norm on horizontal planes.

The sub-Finsler metric in the visualization is defined by the $L^1$ norm on horizontal planes, and is in fact the limit metric of the word metric on the discrete Heisenberg group with standard generators (i.e. $\delta_{1/n} \mathcal{S}^n \to \mathbb{B}_1$ as $n \to \infty$).

Bubble sets are constructed by taking all stable geodesics from the origin to (0,0,1), and are important objects in the study of isoperimetric problems. If you want to know more, read my thesis ;)

Below are some visualizations for the continuous Heisenberg group $H(\mathbb{R})$ that I used in my dissertation work.

In all cases the Heisenberg group is being viewed under exponenti al coordinates, i.e. as $\mathbb{R}^3$ with the group law

$(x_1, y_1, z_1) * (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2 + (1/2)(x_1y_2 - x_2y_1) )$.

Stl files are used for 3d printing. I myself have done so on a Dimension 1200es with some help from the Bray Lab at Tufts.

The metrics under consideration are Carnot Carathéodory metrics defined by norms on the horizontal distribution.

The sub-Riemannian metric is defined by the Euclidean norm on horizontal planes.

The sub-Finsler metric in the visualization is defined by the $L^1$ norm on horizontal planes, and is in fact the limit metric of the word metric on the discrete Heisenberg group with standard generators (i.e. $\delta_{1/n} \mathcal{S}^n \to \mathbb{B}_1$ as $n \to \infty$).

Bubble sets are constructed by taking all stable geodesics from the origin to (0,0,1), and are important objects in the study of isoperimetric problems. If you want to know more, read my thesis ;)

sub-Riemannian Unit Sphere

This can be graphed parametrically as:

$x(t,T) = (\cos(t)(1 - \cos(T)) + \sin(t) \sin(T))/T$

$y(t,T) = (-\sin(t)(1 - \cos(T)) + \cos(t) \sin(T))/T$

$z(t,T) = 2(T - \sin(T))/T^2$

$t \in [0, 2 \pi], T \in [-2 \pi, 2 \pi]$

- Mathematica CDF (with Dilation slider)
- MATLAB FIG
- stl file for 3D printing

This can be graphed parametrically as:

$x(t,T) = (\cos(t)(1 - \cos(T)) + \sin(t) \sin(T))/T$

$y(t,T) = (-\sin(t)(1 - \cos(T)) + \cos(t) \sin(T))/T$

$z(t,T) = 2(T - \sin(T))/T^2$

$t \in [0, 2 \pi], T \in [-2 \pi, 2 \pi]$

Pansu's Bubble Set

This can be graphed parametrically as:

$x(t,T) = (1 - \cos(t))\cos(T) - \sin(T)\sin(t)$

$y(t,T) = (1 - \cos(t))\sin(T) + \cos(T)\sin(t)$

$z(t,T) = (t - \sin(t))/2$

$t \in [0, 2 \pi], T \in [0, 2 \pi]$

- Mathematica CDF (with Dilation slider)
- MATLAB FIG
- stl file for 3D printing

This can be graphed parametrically as:

$x(t,T) = (1 - \cos(t))\cos(T) - \sin(T)\sin(t)$

$y(t,T) = (1 - \cos(t))\sin(T) + \cos(T)\sin(t)$

$z(t,T) = (t - \sin(t))/2$

$t \in [0, 2 \pi], T \in [0, 2 \pi]$

sub-Finsler Unit Sphere

- Mathematica CDF (with Dilation slider)
- MATLAB FIG
- stl fie of sphere of radius 1 for 3D printing
- stl file of sphere of radius 2 for 3D printing

Sphere of Radius 8 in the discrete Heisenberg group with standard generators approximating the CC-sphere

- Manipulable Mathematica CDF

Square Bubble Set

$\Big\{ (x,y,z) \in \mathbb{R} \ \Big| \ |z| \leq \frac{1 - |xy|}{2} , |x|\leq 1, |y| \leq 1 \Big\} $

- Mathematica CDF
- MATLAB FIG
- stl file for 3D printing

$\Big\{ (x,y,z) \in \mathbb{R} \ \Big| \ |z| \leq \frac{1 - |xy|}{2} , |x|\leq 1, |y| \leq 1 \Big\} $