In the case of string theory, consistency requires Spacetime to have 10 dimensions (3D regular space + 1 time + 6D hyperspace).^[1] The Calabi Yau manifold is a complex Fermat Surface with applications in theoretical physics, particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold.^[2]

On the algorithm, Andrew J. Hanson explains the resultant visualisation "contains enough information to check that it’s a consistent local depiction of the complete manifold and so is still a sufficient (although not ideal) representation."

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Tools

• Rhino
• Wolfram Programming Lab

Rhino Plugins

MacOS install paths for:

• Grasshopper: ~/Library/Application Support/McNeel/Rhinoceros/MacPlugIns/Grasshopper/Libraries
• Other: ~/Applications/RhinoWIP.app/Contents/Resources/ManagedPlugIns

Documentation

• Rhino Primer
• mcneel/rhinocommon

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Development

• Install Atom text editor
• Enable Python syntax
• Install the following syntax packages
  • linter-mathematica
  • rhino-python

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Process

• RunPythonScript __init__.py

1. Generate Intersection Curves

• RunPythonScript IntersectSurfaces() to generate intersection curves

2. Generate 2D View

• RunPythonScript Make2d() to generate raw 2D curves.
• Visible PolySurfaces::1
• Visible Intersect::Curves
• Rotate model or select the preferred view
• Select objects on PolySurfaces::1 and Intersect::Curves
• Set DocumentProperties > Model > Absolute tolerance to 0.1
• Run Make2D
• Rename Make2D layer
• Set DocumentProperties > Model > Absolute tolerance to 0.001
• Run Make2D
• Visible Curves::1::0::U
• Select objects on Curves::1::0::U
• Set DocumentProperties > Model > Absolute tolerance to 0.0000000001
• Run Make2d
• Invisible Curves::1::0::U
• Visible Curves::1::0::V
• Select objects on Curves::1::0::V
• Run Make2d
• Perform manual selection/correction using:
  • Trim by one or more cutting objects
  • Split by one or more cutting objects
  • ContinueCurve adding additional Control Points
  • BlendCrv to fill between two visually continuous Curves
  • Join continuous Curves within tolerance of each other

Export/Import Views

• Open source file
• RunPythonScript ExportNamedViews() to write view coordinates to ./views.json
• Create a new file
• RunPythonScript CalabiYau.Run()
• RunPythonScript ImportNamedViews()

Export Vector/Raster

• SelectAll 2D Curves
• File > Export Selected as Adobe Illustrator .ai
• File > Export Selected as Rhino .3dm
• Open exported curves with Adobe Illustrator (NOTE exported SVG data is incompatible with anything other vector editor)
  • Select All
  • Resize selection so that the largest dimension measures 200mm
  • Drag selection into centre of Artboard measuring 210x210mm
  • Set Stroke Width to 0.4mm
  • Save As .eps

Copy A4[SQRDIV].ait to /Applications/Adobe\ Illustrator\ CC\ 2017/Cool\ Extras.localized/en_GB/Templates/A4[SQRDIV].ait

Rasterize File > Export > Export As .psd

• Select whitespace on 5-d-4::visible::lines::PolySurfaces::1
• Create a New Layer at bottom of Layer stack
• Fill selection with paper colour sample

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Sources

1. Supertring Theory
2. Calabi-Yau manifold
3. Wolfram Mathematica [Source]
4. Wolfram Mathematica [Source]
5. Wolfram Mathematica
6. Wolfram Mathematica
7. Wolfram Mathematica
8. Wolfram Mathematica
9. Calabi-Yau and Hanson's surfaces
10. SketchUp
11. Rhino
12. "Visualizing the Path from Fermat's Last Theorem to Calabi-Yau Spaces", Hanson, Andrew J.
13. "Calabi-Yau manifold"
14. "Calabi - Yau Manifolds"
15. "Dimensions of Time"
16. "Mesh Generation with Python"
17. "The Elegant Universe", Greene, Brian
18. "The Fabric of the Cosmos", Greene, Brian
19. "A Construction for Computer Visualization of Certain Complex Curves", Hanson, Andrew J.

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TODO

• Implement concurrency