HANDS ON TROPICAL GEOMETRY

This tutorial requires on OpenSCAD, an open source software for creating solid 3D computer-aided design objects. It is freely available for all platforms on https://openscad.org. The notebook only focuses on the creation of tropical 3D models in OpenSCAD, as the 3D printing process itself depends significantly on the material and the printer used. Size, thickness, and color are easily adjustable in the OpenSCAD file generated by Polymake.

The starting point for creating a 3D-model is one or more polyhedral complexes in $\mathbb{R}^3$. In polymake, objects of type fan::PolyhedralComplex can always be constructed manually by specifying their POINTS and INPUT_POLYTOPES. In special cases, which are covered in the upcoming sections, they can also be constructed via some shortcuts.

Given polyhedral complexes in polymake, the 3D-model can then be created in four steps:

1. Construct a bounding box in polymake. This can be done automatically using our script (which draws a box with a prescribed margin around all vertices) or manually by specifying any bounded polytope (which need not be a cuboid).
2. Export the model from polymake to OpenSCAD using our script.
3. Adjust the thicknesses in OpenSCAD.
4. Export the model from OpenSCAD to any 3D-printable file format.

In the following sections, we will specifically discuss the cases of

1. a single tropical surface,
2. a single tropical curve,
3. a tropical surface containing a tropical curve.

The methods used can however be combined to cover arbitrary arrangements of surfaces and curves.

Before running any of the code below, please load the necessary helper functions and change to application tropical by running the code below

> script("files/hands_on_tropical_geometry/3d_printing_helper_functions.pl");
> application "tropical";

An easy way to construct a tropical surface in $\mathbb{R}^3$ of type fan::PolyhedralComplex in polymake is by constructing and converting a tropical::Hypersurface. This can be done by specifying a tropical polynomial as a string.

> $f = toTropicalPolynomial("min(1+2*w,1+2*x,1+2*y,1+2*z,w+x,w+y,w+z,x+y,x+z,y+z)");
> $tropicalSurfaceTmp = new tropical::Hypersurface<Min>(POLYNOMIAL=>$f);

Important: Since polymake uses homogeneous coordinates, i.e., it identifies polyhedra in $\mathbb{R}^3$ with cones in $\mathbb{R}^4$, tropical polynomials need to be tetravariate and homogeneous instead of trivariate and inhomogeneous.

Alternatively, one can specifying an exponent matrix and a coefficient vector:

> $exponentVectors = [[2,0,0,0], [1,1,0,0], [1,0,1,0], [1,0,0,1], [0,1,1,0], [0,1,0,1], [0,0,1,1], [0,2,0,0], [0,0,2,0], [0,0,0,2]];
> $coefficients = [1,0,0,0,0,0,0,1,1,1];
> $tropicalSurfaceTmp = new Hypersurface<Min>(MONOMIALS=>$exponentVectors, COEFFICIENTS=>$coefficients);

To convert the tropical::Hypersurface to fan::PolyhedralComplex:

> $tropicalSurface = new fan::PolyhedralComplex(
>                      VERTICES=>$tropicalSurfaceTmp->VERTICES->minor(All,~[1]),
>                      MAXIMAL_POLYTOPES=>$tropicalSurfaceTmp->MAXIMAL_POLYTOPES,
>                      INPUT_LINEALITY=>$tropicalSurfaceTmp->LINEALITY_SPACE->minor(All,~[1]));
> $tropicalSurface->VISUAL;

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Constructing a bounding box can be done using the command generateBoundingBox:

> $boundingBox = generateBoundingBox($tropicalSurface);

Note: generateBoundingBox($tropicalSurface) draws a box around all vertices of $tropicalSurface with a margin of 1, generateBoundingBox($tropicalSurface,3,4,5) draw a box around all vertices of $tropicalSurface with margins 3, 4, 5 in x-, y-, z-direction respectively.

or by specifying a custom polytope, which need not be a cuboid:

> $boundingBox = polytope::scale(polytope::cube(3),2);

To construct a bounded polyhedral complex use the command intersectWithBoundingBox, which intersects any fan::PolyhedralComplex with a polytope.

> $tropicalSurfaceBounded = intersectWithBoundingBox($tropicalSurface,$boundingBox);
> $tropicalSurfaceBounded->VISUAL;

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The command generateSCADFileForSurface exports the tropical surface to OpenSCAD. It requires a bounded polyhedral complex and a filename. If the file already exists, it will be overwritten:

> $filename = "foo.scad";
> generateSCADFileForSurface($tropicalSurfaceBounded,$filename);

To solidify the surface into a three-dimensional model, open the exported file in OpenSCAD. Below is a preview of a solidified tropical quadratic surface.

The top of the OpenSCAD file contains parameters which control the thickness of the model amongst other things:

colorSurface = "SlateGray"; // color of surface
scalingFactor = 1; // global scaling factor
thicknessSurface = 0.05; // thickness of surface

1. colorSurface is the color of the surface in the render. It can be specified either by an HTML color name or by RGB value.
2. scalingFactor is a factor by which the polyhedral complex is scaled. It can be used to scale the model to a desired size.
3. thicknessSurface controls the thickness of the surface. OpenSCAD solidifies the surface by replacing every vertex with a ball with specified thickness and taking convex hulls.

Important: In order to 3D-print the tropical surface, every face needs to have a minimal thickness. As a rule of thumb for printing with PLA, we recommend a minimal thickness of 2mm for a print of size 10cm³.

An easy way to construct a tropical curve in $\mathbb{R}^3$ of type fan::PolyhedralComplex in polymake is by intersecting two tropical surfaces. Note however that not all tropical curves can be constructed this way.

Below is an example of how to define a tropical sextic curve as an intersection of a tropical quadratic and a cubic surface.

> # constructing tropical quadric surface
> $mQuadric = [[2,0,0,0], [1,1,0,0], [1,0,1,0], [1,0,0,1],
>                [0,1,1,0], [0,1,0,1], [0,0,1,1], [0,2,0,0],
>                [0,0,2,0], [0,0,0,2]];
> $cQuadric = [1,-1/4,-2/4,-3/4,-3/4,-4/4,-5/4,2/4,0,-2/4];
> $TQuadric = new tropical::Hypersurface<Min>(MONOMIALS=>$mQuadric,
>                                             COEFFICIENTS=>$cQuadric);
> 
> # constructing tropical cubic surface
> $mCubic = [[3,0,0,0], [0,3,0,0], [0,0,3,0], [0,0,0,3],
>            [1,1,1,0], [1,1,0,1], [1,0,1,1], [0,1,1,1],
>            [2,1,0,0], [2,0,1,0], [2,0,0,1], [1,2,0,0],
>            [1,0,2,0], [1,0,0,2], [0,2,1,0], [0,2,0,1],
>            [0,1,2,0], [0,1,0,2], [0,0,2,1], [0,0,1,2]];
> $cCubic = [3,3,3,3,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1];
> $TCubic = new tropical::Hypersurface<Min>(MONOMIALS=>$mCubic,
>                                           COEFFICIENTS=>$cCubic);
> 
> # constructing a tropical sextic curve as the stable intersection
> #   of a quadratic and a cubic surface
> $tropicalCurveTmp = tropical::intersect($TQuadric,$TCubic);

Important: intersect is the intersection of tropical cycles, i.e., it tcomputes the so-called stable intersection, not the set-theoretic intersection.

As before, we convert the tropical curve to type fan::PolyhedralComplex:

> $tropicalCurve = new fan::PolyhedralComplex(
>                    VERTICES=>$tropicalCurveTmp->VERTICES->minor(All,~[1]),
>                    MAXIMAL_POLYTOPES=>$tropicalCurveTmp->MAXIMAL_POLYTOPES,
>                    INPUT_LINEALITY=>$tropicalCurveTmp->LINEALITY_SPACE->minor(All,~[1]));
> $tropicalCurve->VISUAL;

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Constructing a bounding box can be done using the command generateBoundingBox as before:

> $boundingBox = generateBoundingBox($tropicalCurve);
> $tropicalCurveBounded = intersectWithBoundingBox($tropicalCurve,$boundingBox);

For the sake of stability, the tropical curve requires a frame. This is best done using a tropical surface containing it:

> $tropicalSurface = new fan::PolyhedralComplex(
>                      VERTICES=>$TQuadric->VERTICES->minor(All,~[1]),
>                      MAXIMAL_POLYTOPES=>$TQuadric->MAXIMAL_POLYTOPES);
> $tropicalSurfaceFrame = intersectWithBoundingBoxForFraming($tropicalSurface,$boundingBox);
> compose($tropicalCurveBounded->VISUAL,$tropicalSurfaceFrame->VISUAL); 
> #todo: adjust edge thicknesses to distinguish frame from curve

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Exporting the frame and the tropical curve to OpenSCAD can be done using the command generateSCADFileForCurve. It requires two bounded polyhedral complexes and a filename. If the file already exists, it will be overwritten:

> $filename = "foo.scad";
> generateSCADFileForCurve($tropicalSurfaceFrame,$tropicalCurve,$filename);

To solidify the curve into a three-dimensional model, open the exported file in OpenSCAD. Here is the preview of the solidified tropical sextic curve (with frame) constructed above:

As before, the top of the OpenSCAD file contains parameters which control the thickness of the model amongst other things:

colorFrame = "SlateGray"; // color of frame
colorCurve = [0.83,.15,0.27]; // color of curve
scalingFactor = 1; // global scaling factor
thicknessFrame = 0.05; // thickness of frame
thicknessCurve = 0.05; // thickness of curve

1. colorFrame and colorCurve are the colors of the surface in the render. They can be specified either by an HTML color name or by RGB value.
2. scalingFactor is a factor by which the polyhedral complex is scaled. It can be used to scale the model to the desired size.
3. thicknessFrame and thicknessCurve control the thickness of the frame and curve. OpenSCAD solidifies both by replacing every vertex with a ball with diameter equal to the specified radius and taking convex hulls.

The previous section showed an easy way to construct a tropical curve lying on a tropical surface in polymake: start with the surface and construct the curve by intersecting it with another surface. However, as mentioned before, this may not always possible or easy to do. One option that always works is to construct the curve manually by specifying its vertices and edges:

> $linearEquation = toTropicalPolynomial("max(y,z,w)",qw(w x y z));
> $tropicalSurfaceTmp = new tropical::Hypersurface<Max>(POLYNOMIAL=>$linearEquation);
> $tropicalSurface = new fan::PolyhedralComplex(
>                      POINTS=>$tropicalSurfaceTmp->VERTICES->minor(All,~[1]),
>                      INPUT_POLYTOPES=>$tropicalSurfaceTmp->MAXIMAL_POLYTOPES,
>                      INPUT_LINEALITY=>$tropicalSurfaceTmp->LINEALITY_SPACE->minor(All,~[1]));
>         
> $tropicalCurveVertices = 
>    [[1,0,0,0],[1,-8,0,0],[1,-8,-6,0],[1,12,0,0],[0,0,-1,0],[1,-6,-6,0],[1,-11,-3,0],
>     [1,-11,-6,0],[0,-1,0,0],[1,-12,-7,0],[1,4,0,-2],[1,10,0,-2],[1,8,0,-6],[0,0,0,-1],
>     [0,1,1,1],[1,-4,6,6],[1,-4,5,5],[1,-3,5,5],[1,-5,4,4],[1,-3,3,3],[1,-5,3,3]];
> $tropicalCurveEdges = 
>   [[3,4],[0,5],[2,5],[4,5],[1,6],[6,7],[6,8],[2,7],[7,9],[8,9],[4,9],[10,11],[3,11],[11,12],
>    [0,10],[10,12],[12,13],[1,13],[14,15],[8,15],[15,16],[14,17],[16,17],[16,18],[17,19],[8,18],
>    [18,20],[0,19],[19,20],[1,20],[3,14]];
> $tropicalCurve = new fan::PolyhedralComplex(
>                    POINTS=>$tropicalCurveVertices,
>                    INPUT_POLYTOPES=>$tropicalCurveEdges);

(for a preview of the constructed tropical plane with a tropical quartic curve see below)

Constructing a bounding box is easiest done using the command generateBoundingBox as before:

> $boundingBox = generateBoundingBox($tropicalCurve);
> $tropicalSurfaceBounded = intersectWithBoundingBox($tropicalSurface,$boundingBox);
> $tropicalCurveBounded = intersectWithBoundingBox($tropicalCurve,$boundingBox);
> compose($tropicalSurfaceBounded->VISUAL,$tropicalCurveBounded->VISUAL);

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Exporting the tropical surface and curve to OpenSCAD can be done using the command generateSCADFileForSurfaceAndCurve. It requires two bounded polyhedral complexes and a filename. If the file already exists, it will be overwritten:

> $filename = "foo.scad";
> generateSCADFileForSurfaceAndCurve($tropicalSurfaceBounded,$tropicalCurveBounded,$filename);

To solidify the surface and curve into a three-dimensional model, open the exported file in OpenSCAD. Here is a preview of the solidified tropical plane with a tropical quartic curve constructed above:

The top of the OpenSCAD file contains parameters which control the thickness of the model amongst other things:

colorSurface = "SlateGray"; // color of surface
colorCurve = [0.83,.15,0.27]; // color of curve
scalingFactor = 1; // global scaling factor
thicknessSurface = 0.01; // thickness of surface
thicknessCurve = 0.1; // thickness of curve

1. colorSurface and colorCurve are the colors of the surface and curve in the render. It can be specified either by an HTML color name or by RGB value.
2. scalingFactor is a factor by which the polyhedral complex is scaled. It can be used to scale the model to the desired size.
3. thicknessSurface and thicknessCurve control the thickness of the surface and curve. OpenSCAD solidifies the surface by replacing every vertex with a ball with diameter equal to the specified radius and taking convex hulls.

To 3D-print our models using Prusa, we suggest exporting them to 3mf. The advantage of the 3mf file format is that it may contain multiple objects. The relative positions of the object is stored, so that - once loaded into PrusaSlicer - they are aligned as intended and we could assign them differently colored materials. The export of a single 3mf file containing multiple objects can be done with ''%%Colorscad%%''.