Our Catalan Wireframe Polyhedra series is finally complete! Here is the whole happy family of thirteen:

Settings: MakerWare .3mm/low with support-reducing custom slicing profile, on a Replicator 2. We printed each one in a different color with scraps from home and work so that they would be easy to identify. Each model was scaled by a constant that measured the openness of its design, so that together the models would look like a matching set. The wireframe designs were made using Mathematica, MeshLab, and TopMod. Everything in this paragraph is described in detail below.

Thingiverse link: http://www.thingiverse.com/thing:282868

Technical notes, nomenclature flavor: Which polyhedron is which? Every face of a Catalan solid is the same non-regular polygon, and the prefix of its name describes the kind of face that it has.

- Triakis = faces are isosceles triangles arranged in 3-sided pyramids
- Tetrakis = faces are isosceles triangles arranged in 4-sided pyramids
- Pentakis = faces are isosceles triangles arranged in 5-sided pyramids
- Rhombic = faces are rhombi/rhombuses (grammar police line up and fight!)
- Didsyakis = faces are scalene triangles
- Deltoidal = faces are kites
- Pentagonal = faces are irregular pentagons

Technical notes, differentiation and scaling flavors: The following is a key to the Catalan solids pictured above, together with the respective scaling factors as calculated using the algorithm we explained on Day 195:

- BLUE = Triakis Tetrahedron, scale 1.99498 (Day 211)
- BLACK = Triakis Octahedron, scale 2.01059 (Day 202)
- TRANS. ORANGE = Triakis Icosahedron, scale 2.06374 (Day 199)
- NEON ORANGE = Tetrakis Hexahedron, scale 1.83142 (Day 211)
- NEON GREEN = Pentakis Dodecahedron, scale 1.83261 (Day 204)
- GREEN = Rhombic Dodecahedron, scale 1.45648 (Day 210)
- WARM GRAY, scale 1.49535 = Rhombic Triacontahedron (Day 203)
- TRANS. YELLOW = Disdyakis Dodecahedron, scale 1.72338 (Day 195)
- TRANS. CLEAR = Disdyakis Triacontahedron, scale 1.6772 (Day 205)
- RED = Deltoidal Icositetrahedron, scale 1.43823 (Day 198)
- TRANS. PURPLE = Deltoidal Hexecontahedron, scale 1.28527 (Day 196)
- WHITE = Pentagonal Icositetrahedron, scale 1.13082 (Day 197)
- TRANS. BLUE = Pentagonal Hexecontahedron, scale 1.00 (Day 194)

Technical notes, support flavor: Starting from the “Standard PLA” MakerWare .2mm profile with raft and supports, we changed the settings listed below. This is the same knot/wireframe support-reducing profile that we have used many times before, posted here just so you don’t have to go find it:

- “roofThickness”: 0.5,
- “floorThickness”: 0.5,
- “sparseInfillPattern”: “linear”,
- “infillDensity”: 0.2,
- “minSpurLength”: 0.4,
- “doSupport”: true,
- “doSupportUnderBridges”: true,
- “supportDensity”: 0.1,
- “supportExtraDistance”: 0.8,
- “supportModelSpacing”: 0.5,

- Use Mathematica to create the polyhedron and export to STL, and then calculate the scaling factor for the model:

PolyhedronData[“PentakisDodecahedron”]Export[“PentakisDodecahedron.stl”, %]

N[ PolyhedronData[“PentakisDodecahedron”, “EdgeLengths”]]

*Out = {1., 1.12732}*

length = (2*1 + 1.12732)/3

*Out = 1.04244*

edges = PolyhedronData[“PentakisDodecahedron”, “EdgeCount”]

*Out = 90*

surface = N[PolyhedronData[“PentakisDodecahedron”, “SurfaceArea”]]

*Out = 27.9352*scalefactor = Sqrt[(length*edges/surface)]

*Out = 1.83261*

- Use MeshLab to resize and convert to OBJ format:

Import STLFilters –> Normals, Curvature, and Orientation/Scale

to set the scaling factorExport as OBJ

- Use TopMod to remove edges or vertices that don’t belong in the wireframe, and then create the frame and remesh:

Import OBJWireframe 0.250

Remeshing/4-Conversion/Linear Vertex Insertion

Remeshing/4-Conversion/Doo Sabin

Export as STL

- Use MakerWare to size and orient the model:

Import STLAllow MakerWare to rescale or do manually with 2450%

Scale to 50%

Orient model to reduce support

Slice and print with Custom Slicing Profile

UPDATE: This set of models was “Featured” on Thingiverse on April 22, 2014, hooray!

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