The Bohemian Dome

Models X 4 and X 9 are constructed in similar ways - by tracing the points through which a moving circle passes.

The Bohemian Dome

Model X 4 represents a Bohemian dome - a quartic surface with a somewhat convoluted geometric definition. It is helpful to first imagine constructing a torus by taking a large circle and rotating a smaller circle around it, turning it so that the plane through which it passes always meets the centre of the large circle. The Bohemian dome is constucted similarly, but without turning the moving circle (imagine the Earth rotating about the Sun, but not spinning on its axis).

Consider two perpendicular planes $\Pi_{1}, \Pi_{2}$ lying in $\mathbb{R}^3$ (3-dimensional Euclidean space). Consider a circle, $C_{1}$ contained within $\Pi_{1}$. Take another circle $C_{2}$, and rotate it so that its centre always lies on $C_{1}$, and it is always parallel to $\Pi_{2}$. The set of points through which the moving circle passes gives a Bohemian dome. The construction is perhaps best understood when it is seen as well as just explained, so try experimenting with the parameters here to gain insight.

Elements of this construction can be seen in our model: the plane $\Pi_{1}$ containing the stationary circle is vertical (facing out of your screen), and the moving circle stays parallel to the horizontal plane in the image. The highest and lowest positions of the moving circle are visible as the circular "top" and "base" of the model. There are two double lines in the dome, which are seen passing through the centre of the stationary circle $C_{1}$. The dome has rather complicated expressions for its Gaussian curvature and mean curvature [1].

See here for an animation of a nicely decorated Bohemian dome, or here to play with an interactive Bohemian dome.

Model X 4

Model X 4Model X 4Model X 4

 

Model X 9

This model is a degree 8 algebraic surface, generated by moving a circle in a similar (but more complicated!) manner to the Bohemian dome. Consider a unit circle lying in the vertical plane $x=0$ in $\mathbb{R}^3$, with its centre at $(0, 1/2, 0)$. Note that the horizontal diameter of the circle lies in the $z=0$ plane, joining the origin and $(0, 1, 0)$. We move the endpoints of this diameter along the $x$ and $y$ axes, keeping the circle's containing plane vertical, by varying the parameter $t$ such that the horizontal diameter of the circle has endpoints $(t, 0, 0), (0,1-t,  0)$. We then move the circle around similarly three more times in the other three quadrants, with its diameter's motion (click the animation below to see the diameter's path). The animation in fact depicts a Trammel of Archimedes - a tool for drawing circles or ellipses. Observe that from above, the surface resembles the astroid seen when discussing envelopes - this is because as $t$ varies, the diameter follows exactly the same family of lines. The surface self-intersects in two (double) planes, as shown on the model. Less obvious is that the 'top' and 'bottom' of the surface each meet a horizontal plane ($z=1/2$, $z=-1/2$ respectively) in a circle of double points. Note that the model is badly chipped at its four 'corners' - there should be infinitely thin circular arcs along these edges.

 

TrammelModel X 9Model X 9

References

[1] https://www.math.hmc.edu/~gu/math142/mellon/curves_and_surfaces/surfaces...