Modular Origami — Spiky Balls / Stars and Stellated Polyhedra

Models folded and photographed by Michał Kosmulski. Modules designed by their respective authors.
Click on images to enlarge them. Links in image titles lead to pages with more information about each particular object.

Image thumbnails on this page and the full-size images they link to are licensed under a Creative Commons License (Free for non-commercial use with attribution; Ask me about other uses).

Spiked octahedron

Made from Sonobe module (12 modules).

Spiked icosahedron

Made from Nick Robinson's trimodule (30 modules).

Spiked icosahedron

Made from a variant of Sonobe module (30 modules).

Spiked icosahedron

Made from custom units (30 modules).

Spiked dodecahedron

Made from a modified version of Francis Ow's 60 degree module (scroll down the linked page for original unit folding instructions) (30 modules).
[ 3D image ]

Five intersecting tetrahedra (FIT) - 60 degree module

Tom Hull's Five intersecting tetrahedra (FIT)

Made from Francis Ow's 60 degree module (30 modules). The linked page contains some information about both folding the modules and joining them together.
This model is awesome. At only 30 modules it is much more challenging than most models with several times that many units. On Tom Hull's page you can find some tips on joining the modules (this is the fun part of folding this model). Here are some observations I made and which could be useful for other folders:

Spoiler warning: in my case, once I arrived at the rules described below, I was able to finish the model rather quickly. I recommend that you first try it all by yourself, following only the advice found on Tom Hull's page and come back to the instructions below only if you can't manage. Finding out the rules and the deep symmetry of this model are really fun even if it takes more time to finish the model that way.
Start with two frames placed so that a vetex of one goes through the base of the other one and vice versa ("3D star of David").
Look directly at a vertex which goes through the base of the other tetrahedron (e.g. the vertex of the orange tetrahedron closest to the observer in the middle image to the left).
Now, you will have to weave three other tetrahedral frames around the base which consists of two frames. Have a look at the high symmetry of the three additional tetrahedra. One edge of each of them goes under one of the three visible edges of the orange tetrahedron. Two edges of each of these tetrahedra "surround" one edge of the orange tetrahedron placing that edge inside a "fork".
Once you manage to add the third frame (which is the hardest step in my opinion), the fouth and fifth are much easier because due to the high symmetry, you can just copy the third frame, only rotated. Note that under the nearest visible vertex of the orange tetrahedron (and, symmetrically, under every other vertex), three differently colored frames cross, forming a triangle similar to the inner part of the Borromean rings. This structure is partially visible in the last image under the front vertex of the red frame.
Note that the order in which colors appear in each of the pentagonal faces of the underlying dodecahedron is not the same for each face. So for example in the upper image this order is (clockwise): orange, red, dark green, light green, yellow in the face visible in the front but is is different in some of the other faces.
On Tom's page, you will find a tip which says that the finished object has the property that any two tetrahedra are interwoven with one corner poking through a hole of the other and vice versa. This is true and a very useful thing to watch, but keep in mind that this is a necessary condition but not a sufficient one. That means that it is possible to start interweaving the frames and keeping the above condition while creating an intermediate model which will not allow you to add more frames and create the desired finished model. For example, If you add the third frame in such a way that its same vertex pokes through two faces of the two other tetrahedra, you will get a model which fulfills the criterion described above but which can't be extended to create the final model. In the correctly interwoven model, each vertex pokes through exactly one face of exactly one other tetrahedron. A more precise criterion (which I believe to be sufficient) goes: any two tetrahedera A and B out of the five are arranged in such way that:
- One vertex of A pokes through one face of B and vice versa. Each tetrahedron has 4 vertices and each vertex pokes through exactly one face of exactly one other tetrahedron (a different one each).
- The face of A which is opposite to the vertex poking through B lies in a plane parallel to the plane of the face of B opposite to the vertex poking through A.
- Of the 4 faces of tetrahedron A, one is intersected by 3 edges of tetrahedron B (this is the face "poked-through" by B) and three others are intersected by exactly one edge of B each. There are 5 tetrahedrons altogether, and each has 4 faces. Looking at any single tetrahedron, one can say that each of its four faces is poked through by one vertex of a different neighbor. Checking this condition is a good test to see if your model's OK at any given time — for example all arrangements where some face intersects with two edges of the same tetrahedron are deemed to fail.
The frames are interwoven rather tightly. Even with only three of the five frames in place, the frames have very little freedom of movement. If you arrive at a partial model of more than two frames and the frames are loose, moving freely relative to each other, you have probably woven them in an incorrect way.
Have fun! Putting this model together is somewhat mind-boggling and can be frustrating if you only want to finish it as fast as possible. It may take a lot of experimenting to find the right way of weaving the frames but it sure is worth the time.

5 intersecting tetrahedra (FIT) - 60 degree module

Spiked and finned icosahedron

Made from Nick Robinson's trimodule (60 modules).
This is similar to the previous model, but apart from the spikes on all faces, the icosahedron also has "fins" placed on its edges.

Spiked icosahedron

Made from Robert Neale's penultimate module (60 modules).

Spiked pentakis dodecahedron

Made from Sonobe module (60 modules).

There is one spike placed over two adjacent faces of the pentakisdodecahedron in this model. I haven't checked if the angles actually add up, so it might be that this model can only exist because of imprecise folding.