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Stella Tutorial: Paper models of 4D polytopes

With only three dimensions to our world, it can be very hard to visualise four-dimensional objects. Stella4D makes this easier by showing us 3D aspects of these shapes that we can understand, but there's still nothing better than holding a model in your own hands, and 4D can lead to some great 3D models.

Projections

4D polytopes can be projected into 3D, but it's still hard to see what's going on because there's too much clutter obscured inside. It can help to hide some of the cells in order to get a clearer view of what's going on. I'll consider 308-Grit as an example, and describe how I obtained nets for building a paper model.

  • Load the model:
    • "File→Polyhedron List" or Ctrl+N.
    • Type "308" or "grit", hit Enter or click OK.
  • First we want to hide all the truncated cuboctahedron cells, so start by selectng one of them (Shift+Left-click on one of them. They are the blue ones). It should become highlighted in white.
  • Hide all the blue cells ("Display→Show/Hide Cells of Selected Color" or just hit "h").
  • We want full-size cells, so do Ctrl+Shift+Right-click.
  • You should now be looking pretty much at the model we want to make. You may however want to play with the 4D projection to adjust the sizes as you wish. Do this by entering 4D Tumble Mode ( or "Selection→Mouse Selection Modes→4D Tumble Mode"). Then hold down the Space bar and Right-drag until you get the model how you want it. You probably want to adjust the perspective so that the central parts aren't too small, but also so that the outer parts aren't too thin.
  • Click the left-and-down arrow button in the top right corner of the view. It has a small "3" next to it indicating that you will be converting this to a 3D model. You will be asked to choose what to do about the hidden cells, but you will almost always want to just choose the default (top) answer, so just hit Enter.
  • And voila! You have a 3D model that you may treat like any other 3D model. Open an Unfolded Net view to see nets.
19-Tat (the truncated tesseract) is constructed in a similar way, except that I didn't hide all of one type of cell. Instead I hand-picked four of the eight truncated cube cells for hiding. To do this, use Ctrl+h which hides only the selected cell, instead of h which hides all cells of the same colour.

Alternatively, you may wish to build models like 308-Grit by creating each individual projected cell, then gluing them together face-to-face. I used this method for 133-Srip, and Fr. Magnus Wenninger used this method for his 4D models. In this case you need separate nets for each differently-projected cell. So for each cell you will need, perform the following steps:

  • Hide all other cells. Do this as follows:
    • Start with all cells unhidden (Ctrl+Shift+h if necessary).
    • Hide the one cell of interest (select it with Shift+Left-click, then hide it with Ctrl+h).
    • Toggle the hidden cells (Ctrl+Shift+T). This brings back the hidden cell, but hides all others.
  • As before, click the left-and-down arrow button to convert this one cell to a 3D base model. Hit Enter to accept the default response from the dialog box that appears.
Provided you do not change the scale of the original 4D model, the scale for nets of each projected cell should end up matching.

Cross-sections

To see 3D cross-sections of 4D polytopes, use the Cross-Section view. Use Ctrl+Left-drag to change the depth at which the slice is taken. When you get a model you like in the cross-section view, hit the left-and-down button in the top right corner of that view to turn it into a fully-fledged 3D model. Then you can print out nets etc.

Cross-sections where the slicing hyperplane passes through vertices of the polychoron are often particularly interesting, and sometimes a little simpler to make. You can step through these with the yellow left and right arrow buttons at the top of the Cross-Section view, or just by hitting the left or right arrow keys on the keyboard if the Cross-Section view is selected.

Nets

To see the 3D nets of 4D polytopes, just look at the Unfolded Net view with a 4D model loaded. These models are probably most easily made by simply making each cell individually and gluing them together. Be careful to put the cells together in the right way, according to how they appear in Stella4D.

You may wish to use the Cell View (it's the Face View for 3D models). The yellow left and right buttons in the top right corner of the Cell View let you step through the different types of cells. Again, you may use the left-and-down button to convert one of these cells to the base model and print out its nets. In fact, if you set the scale of the polychoron before this ("Scale→Base Polyhedron Edge Length" for example) then this conversion will retain the approperiate scale, so nets for each cell can be printed out in the correct relative sizes.

Tutorials > Paper models of 4D polytopes
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