4D Rendering

• It seems possible to consider this as a broad shader (blu3mo), inspired by Lorentz - Special Relativity in VR.
• Rendering is done based on a 4-dimensional vector that includes time (t, x, y, z).
• How does a regular vertex shader work?
  • It takes (x, y, z) as input and returns (x’, y’, z’).
  • The value of t remains constant.
• 4-dimensional vector:
  • (t, x, y, z) → (tau, x’, y’, z)
  • For all x, y, z, there exist x’, y’, z’.
  • There is only one value of t for all tau.
  • If tau represents real time, we always want a “four-dimensional vector with a certain tau”.
• 
• The challenging part seems to be the rendering.
• We need to have the 4-dimensional data somewhere to render it.
• It is important to consider how to store, edit, and render the data.
• For example, if we store data for all times with fine granularity of delta t, it would be too much data.
• For x, y, z, we use polygons with lower resolution.
• Should we do the same for t?
  • The concept of a 4D polygon is interesting.
• Existing examples:
  • All use x, y, z, w instead of x, y, z, t.
  • However, the tasks should be the same.
  • 4-polytope - Wikipedia
  • GitHub - mwalczyk/four: 💎 A 4-dimensional renderer.
  • Marc Ten Bosch
    • Miegakure Hide and Reveal: A 4D puzzle-platforming game.
    • N-dimensional rigid body dynamics | ACM Transactions on Graphics
    • No discussion about rendering.
    • https://www.vice.com/en/article/9kmknd/what-happened-to-miegakure-the-game-that-promised-the-4th-dimension
    • A look at the Technology behind the 4D Game Miegakure « mtbdesignworks Blog Miegakure, 4D Toys, Marc ten Bosch
      • Uses tetrahedrons instead of triangulation.
  • https://replit.com/talk/share/FOUR-DIMENSIONAL-rendering-engine-in-WebGL/139089
    • Uses raymarching for 4D rendering.
    • Becomes pixelated due to computational complexity.
  • https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1921&context=cstech
    • Contains detailed discussion.

    • Generating and Rendering Four-Dimensional Polytopes

  • http://honors.cs.umd.edu/reports/AMartynov/4D_RayTracer.htm
    • Uses sphere space.
    • Similar to the previous example.
  • Engine 4 | Modeling | Unity Asset Store
    • Development seems to have ended.
  • GitHub - Jellevermandere/4D-Raymarching: a Unity framework to create spacial 4 dimentional games, using raymarching
    • Custom implementation of raymarching.
    • Low resolution to maintain performance with many objects.
  • 4D Golf Tech Details
    • Understanding 4D tetrahedra.
    • Slicing reveals triangles (not limited to triangles) as output.
    • Obtaining a 3D triangle by slicing a 4D tetrahedron with four (x, y, z, t) values.
• It seems that we also need to reimplement physics in 4D.
• Before dealing with relativity, classical mechanics needs to be implemented since rigid bodies cannot be used.
• Interesting!
• Options:
  • https://replit.com/talk/share/FOUR-DIMENSIONAL-rendering-engine-in-WebGL/139089
  • https://github.com/Jellevermandere/4D-Raymarching
  • Both are pixelated, reaching the performance limit.
  • Unlike 3D to 2D, only one slice is needed.
  • Therefore, ray-marching for the entire volume is computationally inefficient.
  • Convert 4D polygons to 3D polygons.
  • No open-source code found.
• 4D Physics.- If something is changed at t=1, it is necessary to compute everything that follows.
  • That’s impossible.
• Some simplified physics?
• In other words, it seems like there could be more interesting things here (blu3mo).
  • The important point is causality.
    • Allowing causality in directions other than time would make rendering too difficult.