Scenes for a study of scale and dimension

[camden-bock]

Is your feature request related to a problem? Please describe. This is discussed in Dimmel's proposal, appendix B. MODIFIED AUG 19, 2019

Describe the solution you'd like A set of figures that extend the sandbox dimension with analogs in 2,3 and 4 dimensions.

The scene will have a set of figures that can be viewed using a set of lenses. Additionally, some actions (events) can be applied to each figure).

Figures:

• 1-,2-,3-sphere
• square, cube, 8-cell
• triangle, tetrahedron, 5-cell
• 1-,2-,3-,4-dimensional Cartesian axes (as lines or basis vectors)

Lenses

• Standard Rendering (not available for 3-sphere, 8-cell, 5-cell, 4-d axes)
• Parallel Projection
• Projective Projection
• Stereographic Projection (inflate to n-sphere)

• Cross-Section

  Events

• Translation
• Rotation

• Net folding/unfolding

  Completion Checklist

Projection Algorithms

• [ ] Parallel Projection
  • [ ] 4D --> 3D
  • [ ] 3D --> 2D
  • [ ] 2D --> 1D
• [ ] Projective Projection
  • [ ] 4D --> 3D
  • [ ] 3D --> 2D
  • [ ] 2D --> 1D
• [ ] Stereographic Projection (inflate to n-sphere)
  • [ ] 4D --> 3D
  • [ ] 3D --> 2D
  • [ ] 2D --> 1D

Cross-Sections --> Are these a separate object?

• [x] Sphere
  • [x] 1-Sphere
  • [x] 2-Sphere
  • [x] 3-Sphere
• [ ] Cubes ???
  • [ ] Square
  • [ ] Cube
  • [ ] 8-cell
• [ ] Tetrahedrons ???
  • [ ] Triangle
  • [ ] Tetrahedron
  • [ ] 5-cell
• [ ] Axes ???
  • [ ] 1D
  • [ ] 2D
  • [ ] 3D
  • [ ] 4D

Net Folding --> These can exist if we consider an unfolding of the mesh

• [ ] Sphere ???
  • [ ] 1-Sphere
  • [ ] 2-Sphere
  • [ ] 3-Sphere
• [ ] Cubes
  • [x] Square
  • [x] Cube
  • [x] 8-cell
• [ ] Tetrahedrons
  • [x] Triangle
  • [x] Tetrahedron
  • [x] 5-cell
• [ ] Axes ???
  • [ ] 1D
  • [ ] 2D
  • [ ] 3D
  • [ ] 4D

Translation

• [x] Sphere
  • [x] 1-Sphere
  • [x] 2-Sphere
  • [x] 3-Sphere
• [x] Cubes
  • [x] Square
  • [x] Cube
  • [x] 8-cell
• [x] Tetrahedrons
  • [x] Triangle
  • [x] Tetrahedron
  • [x] 5-cell
• [ ] Axes
  • [ ] 1D
  • [ ] 2D
  • [ ] 3D
  • [ ] 4D

Rotation

• [x] Sphere
  • [x] 1-Sphere
  • [x] 2-Sphere
  • [x] 3-Sphere
• [x] Cubes
  • [x] Square
  • [x] Cube
  • [x] 8-cell
• [x] Tetrahedrons
  • [x] Triangle
  • [x] Tetrahedron
  • [x] 5-cell
• [ ] Axes
  • [ ] 1D
  • [ ] 2D
  • [ ] 3D
  • [ ] 4D

Important Design Considerations If we allow translation and rotation, do we need more general cross-sections? Can we solve all of the projection problems with 4D --> 3D?

[camden-bock]

@reneyost123 is working on commenting the NetFoldingTasks, and checking their accuracy.

[camden-bock]

@camden-bock added network support through PUN, with a build flag toggle.

In theory, an avatar system can be dropped in place. If the slider is synced, no other values need to be synced. The avatar system can be taken out of the HigherDimDodgeball scene.

[camden-bock]

The slider needs to be debugged and bounded. There should be variables in the editor for the bounds fo the slider. (consider intrepreting slider values from 0 to 1 and using a Vector2 to set max/min bounds.)

[camden-bock]

UV mappings need to be done for hypercube and tetrahedron

[camden-bock]

Double-sided textures need to be applied.

[camden-bock]

@reneyost123 has been developing a script for copying a mesh at runtime for task set 3. We then need to have mesh inputs for each of these figures.

All of these figures could be considered cases of the figures in Task 1, where they are completely folded. Alternatively, we could export blender models for each figure.

[camden-bock]

We need to add editor controls for changing the perspective on 4D objects.

[camden-bock]

Added 4D perspective controls. Attempted to patch slider bugs. Needs testing

[camden-bock]

We need to add line renders to the 4D objects to show their 2D edges. @reneyost123 please take a look at this this week.

[camden-bock]

@reneyost123 has added a script that copies and scales any mesh and line render on an object, which dynamically updates.

[camden-bock]

The five cell is pretty confusing to look at.

[camden-bock]

function key mapping for toggling objects added.

[camden-bock]

There are some weird mathematical errors in the folding script. We might consider moving to Unity.Mathematics.

[camden-bock]

Both the five cell and the eight cell are not folding properly. This is clear when we rotate the representation.

Part of the problem might be using 4D euler angles. 3D euler angles don't work well, so I'd imagine that there are more problems in 4D.

It seems that rotors might be the generalization we are looking for. @reneyost123 let's think about how to write a general case for this (R^n) based on the float[n] construct in Unity.Mathematics. We would write a static script that we would use throughout our project.

Similarly, we should rewrite our projection algorithm to be generalized for R^n. The two of these will take some linear algebra and some algebra to sort out.

[camden-bock]

Alternatively, the new Unity math system supports matrices. We could do a similar process with a matrix.

https://math.stackexchange.com/questions/197772/generalized-rotation-matrix-in-n-dimensional-space-around-n-2-unit-vector

[reneyost123]

The net folding for the triangle, square, tetrahedron, and cube is working properly both with the override option for folding and the animated folding.

The five-cell and hypercube still need to be worked on.

[reneyost123]

The cross-section of a torus could result in a few shapes, such as an ellipse, or two circles.

https://en.wikipedia.org/wiki/Spiric_section

[reneyost123]

The mathematics for a spiric cross-section has been solved for and parameterized, though certain constant parameter values need to be decided on.

[camden-bock]

Prototype Status:

• [ ] Net Folding Tasks 2D

  • [x] Condition 1 net: four line segments --> square
  • [x] Condition 2 net: three line segments --> equilateral triangle 3D
  • [x] Condition 1 net: six squares --> cube
  • [x] Condition 2 net: four equilateral triangles --> tetrahedron 4D
  • [] Condition 1 net: eight cubes --> hypercube
  • [ ] Condition 2 net: five tetrahedrons --> hypertetrahedron

• [ ] Cross Section Tasks [For this set of tasks, participants will be able to move an (n-1)-dimensional slicer across a hidden n-dimensional figure.] 2D

  • [x] Condition 1 figure: a circle
  • [x] Condition 2 figure: an annulus 3D
  • [x] Condition 1 surface: sphere
  • [ ] Condition 2 surface: torus 4D
  • [ ] Condition 1 surface: hypersphere (cross sections will be spheres of different sizes)
  • [ ] Condition 2 surface: hypertorus (cross sections will range from ellipses to pairs of toruses)

• [ ] Measure Tasks 2D

  • [x] Condition 1 similar figures: square, larger square
  • [x] Condition 2 similar figures: equilateral triangle, larger equilateral triangle 3D
  • [x] Condition 1 similar figures: tetrahedron, larger tetrahedron
  • [x] Condition 2 similar figures: cube, larger cube 4D
  • [ ] Condition 1 similar figure: hypercube, larger hypercube: the figures will be shown in perspective as 3D projections
  • [ ] Condition 2 similar figures: hypertetrahedron, larger hypertetrahedron: the figures will be shown in perspective as 3D projections

[camden-bock]

The toric sections need some work to debug.

[camden-bock]

Hypersphere math is done. Used this result:

https://math.stackexchange.com/questions/1159613/would-the-cross-section-of-a-hypersphere-be-a-sphere

[reneyost123]

Points have been added to the net foldings of the square and triangle. Points have also replaced line segments for cross-sections of 2-dimensional objects.

[camden-bock]

Hypertoric sections source: http://www.dr-mikes-maths.com/4d-torus.html

[reneyost123]

Trying to solve for the maximum value of theta to find the location to start the sweep for the torus cross-section. After solving for r, we set both positive solutions equal to each other and solved for theta.

[camden-bock]

Most of the figures are prototyped:

We continue to have problems on the toric sections. That is on hold for now, considering replacing with conic sections.

The regular 5-cell's dihedral angle needs to be corrected.

[camden-bock]

Tweak all 4D nets and measures for better projections.

[camden-bock]

Previous Design Spec

Is your feature request related to a problem? Please describe. This is discussed in Dimmel's proposal, appendix B.

Describe the solution you'd like A set of figures that extend the sandbox dimension with analogs in 2,3 and 4 dimensions.

Enhancements

• [ ] Lit line renderers (similar to tilt brush)
• [x] animate folding and unfolding
• [x] scalable figures (micro vs meso)
• [x] orthogonal vs prospective perspectives?
• [x] Individually color faces of nets
• [ ] stereo-graphic projection?

Additional context

• [x] Net Folding Tasks 2D

  • [x] Condition 1 net: four line segments --> square
  • [x] Condition 2 net: three line segments --> equilateral triangle 3D
  • [x] Condition 1 net: six squares --> cube
  • [x] Condition 2 net: four equilateral triangles --> tetrahedron 4D
  • [x] Condition 1 net: eight cubes --> hypercube
  • [x] Condition 2 net: five tetrahedrons --> hypertetrahedron

• [ ] Cross Section Tasks [For this set of tasks, participants will be able to move an (n-1)-dimensional slicer across a hidden n-dimensional figure.] 2D

  • [x] Condition 1 figure: a circle
  • [x] Condition 2 figure: an annulus 3D
  • [x] Condition 1 surface: sphere
  • [ ] Condition 2 surface: torus 4D
  • [x] Condition 1 surface: hypersphere (cross sections will be spheres of different sizes)
  • [ ] Condition 2 surface: hypertorus (cross sections will range from ellipses to pairs of toruses)

• [x] Measure Tasks 2D

  • [x] Condition 1 similar figures: square, larger square
  • [x] Condition 2 similar figures: equilateral triangle, larger equilateral triangle 3D
  • [x] Condition 1 similar figures: tetrahedron, larger tetrahedron
  • [x] Condition 2 similar figures: cube, larger cube 4D
  • [x] Condition 1 similar figure: hypercube, larger hypercube: the figures will be shown in perspective as 3D projections
  • [x] Condition 2 similar figures: hypertetrahedron, larger hypertetrahedron: the figures will be shown in perspective as 3D projections

[camden-bock]

All perspective data for projections should be formed into a struct.

[reneyost123]

These pictures generally outline the mathematical process for calculating the cross section of a plane and a triangle.

[camden-bock]

The class for this structures the lenses into four objects:

Net mesh Standard Mesh (can be RTS, projected) Cross-section Mesh

Everything needs to be rendered with mesh renderers (no line renderers).

[camden-bock]

Actions are generally RTS at the world level. Everything in handled for 4D, though only 3D may be relevant.

[camden-bock]

Move to a different repository.