Planar Sections of Triply Periodic Surfaces

by Roberto De Leo

This page contains numerical data on the topology of planar sections relative to several triply periodic functions. Our main interest in the subject comes from the following result:

Theorem (Zorich 1984, Dynnikov 1999).

Given a generic smooth triply periodic function $F(x,y,z)$:

As a consequence of this result, given a level set $F_c=\{F(x,y,z)=c\}$ of a triply periodic function $F$, the set of all directions $\mathbf B$ is sorted in "islands" so that all directions belonging to the same island share the same vector $\mathbf u$. These islands are surrounded by a "sea" of directions that give rise to only closed orbits. Such subdivision in islands is called Stereographic Map of a surface.

Finally, since Stereographic Maps of two level sets of the same function are "compatible", it is possible to define also the Stereographic Map of a function. In this case there is "white sea" because for any direction $\mathbf B$ there is a level set $F_c$ on which open intersections arise. Below you can see some of the Stereographic Maps that have been found numerically so far.

Already in literature

$F(x,y,z)=\cos x+\cos y+\cos z$



This function is one of the simplest trigonometric polynomials in 3 variables having some of its level sets connected. Each of these connected level sets is a triply periodic surface of genus 3. The model in the left shows two periods in $x$, $y$ and $z$ of the level set $F_0$.

This function has the following property: the level sets $F_c$ and $F_{-c}$ are a translation of each other, so that the level $F=0$ has the property: for every vector $\mathbf B$ there is some plane perpendicular to it having open intersection with $F_0$. Hence we can get the Stereographic Map associated to $F$ by looking at a single level, namely $F_0$.

Finally, notice that $F_0$ is a good approximation to the triply periodic minimal surface called Schwarz' P Surface.

References:

$G(x,y,z)=\mathrm{plcos\ } x+\mathrm{plcos\ } y+\mathrm{plcos\ } z$

The function $\mathrm{plcos\ } x$ is the simplest piecewise linear approximation of $\cos(2\pi x)$ (its graph is shown on the right). The level sets of $G$ have the same symmetry discussed above for $F$ and so even in this case it is enough to study the zero level set in order to study the stereographic map of the whole function.

The 3D model on the left is the level set $G_0$ is called muoctahedron and it is one of the 3 regular skew apeirohedra of the Euclidean 3-space.

References:

$H(x,y,z)=\mathrm{mid\ }\{|2x-1|,|2y-1|,|2z-1|\}-\frac{1}{2}$

This piecewise linear function cannot be written as the sum of three functions in one variable but features the same symmetry as the previous two functions. In particular, its stereographic map is the same as the stereographic map of its level set $H_0$.

This level set, whose 3D model is shown on the left, is called mucube and it is another one of the 3 regular skew apeirohedra of the Euclidean 3-space.

It is remarkable that the stereographic map for this particular case can be described by a simple recursive algorithm. This made possible, on one side, to test to a very high degree of accuracy the numerical results of the NTC software, used to produce all these numerical data, and, on the other side, to prove a conjecture of Novikov that the measure of the set of points $R$ not contained in any "island" is zero. Recently also the second part of the conjecture was also proved rigorously, namely that the Hausdorff dimension of $R$ is striclty between 1 and 2.

References:

Work in progress

The cubic polyhedron $P_1$

This polyhedron is the unique cubic polyhedron with all screw vertices whose type $S$ and $Z$ alternate in a checkerboard fashion (see Cubic Polyhedra), just like the mucube above is the only cubic polyhedra with all monkey saddles.

Like the previous three surfaces $F_0$, $G_0$ and $H_0$, this surface has the property that its interior is a translation of its exterior, which ultimately implies that planes perpendicular to any direction cut this surface into some open intersection. Hence, any function having this surface as a level set has a Stereographic map equal to the one of this very surface.

Unlike the surface above, though, this one is not symmetric by the exchange of the three coordinate axes and so neither is its corresponding Stereographic Map.

A family of icosahedra

This 2-parametric family foliates the 3-torus so that every leaf is either a single point or a icosahedron with 8 triangular and 12 square faces. The "holes" are all squares. Every leaf is a genus-3 surface.

As above, this function's Stereographic Map cannot be obtained as the SM of a single level set.

$F(x,y,z)=\cos x\cos y+\cos y\cos z+\cos z\cos x$

This function has the property that all of its connected level sets have genus 4, namely they are topologically equivalent to a sphere with 4 handles attached.

Unlike all previous cases, this function's Stereographic Map cannot be obtained as the SM of a single level set.

The family of polyhedra $\mu G4$

This family is obtained by taking a truncated octahedron (see the function $G(x,y,z)$ above) and replacing each of its hexagonal faces by a cylinder. This way we get a (singular) foliation of the 3-torus into genus-4 polyhedra.

Just as the previous case, for this family the Stereographic Map cannot be obtained as the SM of a single surface.

$F(x,y,z)=\cos x+\cos y+\cos z + \cos x\,\sin y\,\sin z + \sin x\,\cos y\,\sin z + \sin x\,\sin y\,\cos z$

The surfaces of this family have the same kind of symmetries as $\cos x+\cos y+\cos z$, namely the stereographic map of the surface at the level $0$ coincides with the stereographic map of the whole function. Every regular surface of the family has either genus 0 or genus 3.

To Do list...

The family of polyhedra $\mu G3$

This family is obtained by taking a truncated octahedron (see the function $G(x,y,z)$ above) and replacing each of its square faces by a cylinder. This way we get a (singular) foliation of the 3-torus into genus-3 polyhedra.

The family of polyhedra $\mu G7$

This family is obtained by taking a truncated octahedron (see the function $G(x,y,z)$ above) and replacing each of its faces by a cylinder. This way we get a (singular) foliation of the 3-torus into genus-7 polyhedra.

The family of piecewise smooth surfaces S3

The surface $S3_R$ of this family is obtained as the boundary of the union of the balls of radius $R$ centered at all points with integer coordinates.

The family of piecewise smooth surfaces S4

The surface $S4_R$ of this family is obtained as the boundary of the union of the balls of radius $R$ centered at all points with integer coordinates and all points with all half-integers coordinates.

The family of piecewise smooth surfaces ST3

The surface $ST3_R$ of this family is obtained as the boundary of the union of the balls of radius $R$ centered at all points with integer coordinates and the cilinders with axes the coordinate axes, centered at the points with all half-integer coordinates and with radius $R\sqrt{1-R^2}$.

The family of piecewise smooth surfaces ST4

The surface $ST4_R$ of this family is obtained as the boundary of the union of the balls of radius $R$ centered at all points with integer coordinates and with all half-integer coordinates and the cilinders with axes the diagonal of the cube, centered at the points with all half-integer coordinates and with radius $\sqrt{R^2-\rho^2/3}$, where $\rho=R(\sqrt{3}+2(\sqrt{2}-\sqrt{3})R)$.

$G(x,y,z)=\alpha(\cos x+\cos y+\cos z)+\cos(\beta x)+\cos(\beta y)+\cos(\beta z)$

The surfaces of this family have the same kind of symmetries as $\cos x+\cos y+\cos z$, namely the stereographic map of the surface at the level $0$ coincides with the stereographic map of the whole function. Every regular surface of the family has either genus 0 or genus 3.