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This screen-shot is from
Small Stella, but could equally well
be from Great Stella or
Stella4D
(except the toolbars would look a little different).
It shows a three-view layout. There are a number of layouts to choose
from, showing anywhere between one and six views at a time. Each view
may be one of many different types. Here we can see a view of a
uniform polyhedron (the icosidodecahedron in this case), a view of the
2D net required to build it, and a 3D view of the net folding up.
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Here we see a single-view layout, showing a view where the net may be
folded up into the model using the mouse.
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Images may be put on faces of polyhedra now. They will also appear on
nets when printed out. Here a photo of my cat has been put on the
pentagonal faces. This screen-shot also shows spheres at the vertices
and cylinders along the edges, in a wooden finish.
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Small Stella,
Great Stella and
Stella4D can all display the
rotational and reflective symmetries of any polyhedron.
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| Cross-sections of various polyhedra |
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Here we see cross-sections of various polyhedra. The animation comes
from moving the slicing plane through the model over time, and can be
shown in real-time within the software
(Small Stella,
Great Stella or
Stella4D).
Top left shows cross-sections embedded in the cube. This helps to show
how these slices fit into the original model. Can you tell which
polyhedra are being sliced in the other images?
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A screen-shot from the Demo version of
Great Stella.
This shows the Rainbow colouring mode, and the fact that the
icosahedron may be stellated in the Demo
version.
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This is a screen-shot from
Great Stella,
showing the stellation diagram attached to the face of a 3D model (the
icosahedron). The lines indicate
where other face planes intersect with this one. You can click on a
2D region of the diagram to enable the 3D cell behind that region,
as shown here.
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This screen-shot is from Small Stella
(and would also be available in
Great Stella and
Stella4D of course).
It shows a single-view layout, with 3D stereo enabled. You need to
wear red-blue or red-green glasses to view this image properly. Click
on the image for a full size version.
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This screen-shot is from
Great Stella.
It shows a five-view layout. The only view here that wouldn't be
available in
Small Stella
is the one in the bottom right corner, which shows the stellation
diagram for the great icosahedron
in the big view. The model itself is available in
Small Stella.
Click image to see it full size.
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Another screen-shot from
Great Stella,
showing the small inverted retrosnub icosicosidodecahedron (aka Yog
Sothoth). This model is only available in
Great Stella and
Stella4D.
Click image to see it full size.
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Use this toolbar button to enter "Create Facets Mode"...
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...then use Shift+Left-click on the vertices of the
facets you want, which will become faces of a new model when you
use "Poly>Create Faceted Polyhedron". In this example we
create a compound of 15 cuboids. Only three facets need to be created
manually because they will be repeated over the symmetry group to
create the complete set of faces. Finally we colour them in orthogonal
sets of three using "Color>Special Color Arrangements>Rhombic
Triacontahedral Arr 1 (5 colors)".
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This shows the augmentation feature of
Great Stella. You can augment any
model with any other model. Here a pentagonal pyramid (J2) is being
augmented with a square pyramid (J1) at one of the triangular faces.
The user sees a preview of the result and can cycle through the
possible orientations before choosing the one they want.
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Overlapping coplanar faces have their colours blended in real-time.
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Below are three animations showing views where a uniform polyhedron
can be morphed into its dual. Each shows a different model, and uses a
different method for doing the transformation. In the software, the
transformation is controlled interactively with the mouse. See
my paper for a description of
these techniques.
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Here is the best known polychoron and probably the simplest to understand.
It is the tesseract, also known as the 8-cell or 4D cube. It has eight
cubes as its cells (or sides). Here it is shown with all the cells
hidden, leaving just the edges and vertices visible. It has been projected
into 3D so that we may view it in our world, just as a 3D model is
projected into 2D when we view it on a computer screen.
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Getting a bit more complicated, here is a cantitruncated tesseract (uniform
polychoron number 308, aka "grit", in Stella4D). This polytope has 56
cells. 32 truncated cuboctahedra (hidden),
16 truncated tetrahedra (yellow), and 8
triangular prisms (red). Again, it has been projected into 3D, this time
with only the truncated cuboctahedral cells hidden.
See my paper model here.
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Each "side" of a 4D polytope is a 3D polyhedron, just as each side of a
3D polyhedron is a 2D polygon; and just as in 3D, you can unfold the sides
into a "flat" net. I put "flat" in quotes because here flat means that the
unfolded cells lose their four-dimensionality, being reduced to 3D. Here
you see two such nets. The first for the truncated tesseract (19-Tat in
Stella4D). The second is for the dual of another uniform polychoron
(20-Thex).
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Here is the tesseract rotating in 4D before being projected into 3D.
Stella4D lets you rotate like this
interactively.
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The tesseract again, with a cell-first cross-section passing through it.
The cross-section itself doesn't change, remaining a cube throughout, but
appears to distort due to the projection from 4D into 3D.
Stella4D lets you interactively change
the slicing depth with the mouse.
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This time it's a vertex-first cross-section, which starts as a growing
tetrahedron, gets truncated towards an octahedron, then extends out the
other way back into a tetrahedron. Again, it appears distorted throughout
due to the perspective projection from 4D into 3D.
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Here are cross-sections of the 600-cell, which has 600
tetrahedra as its sides. Not embedded in
the original 4D model here, so no distortion this time.
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Getting more complex now, and we haven't even gone beyond the regular
polychora yet! Here we see cross-sections of the great grand stellated
120-cell (16-Gogishi in Stella4D), which
has 120 great stellated dodecahedra as its
cells.
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Stella Screen-Shots
