|The 120 Cell is a 4D structure made
regular dodecahedra. This "shadow" of it has the form of one
filled in with 119 smaller dodecahedra. In 4D all the
but in this 3D shadow, angles are necessarily distorted, so
and outermost dodecahedra appear regular.
At right is shown an SLS model, about 3 inches in diameter, made on a DTM 2500Plus machine. It is a beautiful object to hold in your hand and rotate slowly. To make your own, just download this STL file (0.35MB) and send it to your local RP machine.
Below is a gorgeous 4-inch model made by the Extrude Hone "ProMetal" process. Made of stainless steel powder which is infiltrated with bronze to braze it together, it should be quite long lasting. So I am happy to think that in 2000 years, someone could be holding this same model in their hands and spinning it around as I am now. These are available for purchase through Bathsheba Grossman.
|Here is a popular fractal, often called the
Tetrahedron, because it is a three-dimensional
the two-dimensional Sierpinski triangle. This is a
version, as there are five different sizes of octahedral
model is scaled so the tetrahedron edge is about 8.5
inches long. You
can get a better sense of scale from the image at the top
of this page.
It is easy to write a program which generates a computer-rendered image of this form, but it usually has the little solid tetrahedral parts just barely meeting at mathematical points. If you designed a physical model like that, it would fall apart into dust. An RP machine requires a well connected solid interior (with a triangulated manifold boundary) so the software to produce this file has a few tricky aspects.
Here is the stl file (1.0 MB) for you to download and feed to your own RP machine, to make your own model.
|Even more beautiful and intricate is the truncated
a 4D object made of 120 truncated dodecahedra and 600
is shown an SLA model of an orthogonal shadow of it, about
diameter. It is quite stunning to view the tunnels which
six different directions. To make your own, just download this STL file (0.81 MB) to
your local RP machine.
The paper cited above also describes this structure.
I used these and many other 4D shadow models in a seminar on The Fourth Dimension which I taught at Stony Brook University in the spring of 2003.
You may also be interested in reading a paper describing the mathematics behind this and other models of 4D objects I have made.
|This is a model of a polyhedron
first described by the mathematician Michael Goldberg in a
is "8,3" in his series of such polyhedra. For this
model, I chose
largest one with under 1000 faces; it has 972 faces---12
hexagons.) But as far as I know, neither he nor anyone else
years had previously worked out how to calculate the lengths
make these polyhedra with planar faces. I'll describe more
and ways to make them in a future paper.
This is a five-inch diameter model made on a 3D Systems InVision machine. If you want to make your own, here is the STL file (1.7MB).
|Here is a nested series of seven spheres
can rotate freely, independently of each other. This
construction is an
homage to the long tradition of turning concentric ivory
spheres on a
lathe. As outlined on this
page, this artistic tradition started in Nuremburg in
seventeenth century and is still carried out in parts of
Each sphere is based on the edges of a different Goldberg polyhedron. From inside to inside, they are:
When built, the twelve pentagons are aligned in all the spheres, so you can look right through them in to the center and out the other side. After randomizing the orientations, it is something of a puzzle to restore this property.
|This is a tangle of ten
equilateral triangles, just one of many interesting polygon
first described by Alan Holden. The stl file is here (0.01 MB).
For the history of these forms and many more examples, see this paper. The bottom of that page also has a link for you to get the java software which I wrote that allows you to generate the stl files for all of Holden's tangles plus infinitely many new ones.
|Here are models of two
polyhedral compounds with icosahedral symmetry. These were
described in the mathematics literature by John Skilling in
of anyone making a physical model of either of them before I
At top is the compound of five concentric truncated tetrahedra. Below is the compound of six concentric pentagonal prisms. To understand these, you must see each as several interpenetrating solids. In the compound of five truncated tetrahedra illustrated at top, you can see a large equilateral triangle facing you. Its edge length is roughly the same as the overall radius of the object. There are also large hexagons of the same edge length. One truncated tetrahedron consists of four of these triangles and four of the hexagons. The overall form is five of these interlocked. Similarly, in the lower figure find large squares and pentagons of the same edge length. Five squares and two pentagons make one pentagonal prism, and there are six interlocked prisms. It may be easier to see the structure if you read the stl file into a 3D viewing program and slowly rotate the view.
The models are made of plaster on a Zcorp machine, which uses inkjet printer technology to squirt water selectively in the places where the plaster dust is to be hardened, leaving the unmoistened plaster dust to be vacuumed away. The stl files are here (0.03 MB) and here (0.03) MB.
|Here is another popular
the Menger sponge. The FDM model shown at right is
there are three sizes of holes. The stl file to make your
own is here (1.8 MB).
The surface area (and therefore file size) grows exponentially with the order. If you would like me to email you the 26 MB stl file for a fourth-order Menger sponge, let me know. It is easier for you to generate it on your own computer. A simple algorithm can be based on the idea that a voxel is empty if there exists a trit position (in base three representation) where two or more of its X, Y, and Z coordinates have the value one. If you have access to Mathematica, you can use my software in this paper to generate stl files for Menger sponges of any order.
The lower image at right shows two halves of a third-order Menger sponge, with a hexagon of paper between them. You probably know how to cut a cube in half with a planar slice to create a hexagonal cross section: you cut on a plane which is a perpendicular bisector of any of the cube's four long diagonals. (If you don't understand this, go cut a cube of cheese or potato, then come back.) A great visualization exercise is to figure out what the cross section is when you cut a Menger sponge along that hexagonal slicing plane. In other words, what is shape of the area where the half-sponge is in contact with the paper?
The stl file for you to build your own halves is here (1.1 MB). The model shown at right has 5.5 cm edge length, made by selective laser sintering. It is a fun puzzle to give someone the two halves together and ask them to try to visualize and draw the boundary surface before separating the halves. Then they can open it up and see how close they came to the correct cross section.
To give you a chance to think about it before seeing the answer, I put a photo of the separated halves on a separate page, here.
|These are tetrahedra and an octahedron built from a small inventory of block shapes that are Voroni cells around face-centered cubic lattice points in the polyhedra. A more detailed explanation (and the stl files to make your own set) are available here.|
|Here is a 4-inch model of
sphere. There are 260 triangles in the outer layer. The
layer has 12 pentagons and 120 hexagons. As far as I know,
it is the
world's only chiral two-layer geodesic sphere, as explained
The stl file, if you want to make your own copy on your own RP machine, is 20MB.
|This is an old favorite of
the snub truncated icosahedron. The pentagons and
hexagons of a
soccer ball are separatged by a chiral boundary of
Read more about it here.
|Here is a fairly cool object,
you like music and geometry: It is a model of the orbifold
three-note chord types, as described by Clifton Callender,
and Dmitri Tymoczko here.
chord, but abstracting away
particular transposition. The struts connect pairs of
one voice changing one semitone. The top left is the
chord, sitting at the cone point. It is adjacent to the
major triad and
the minor triad. At the bottom right is the "unison
meaning all three voices sound the same pitch. Most chords
neighbors, because any of the three pitches can be raised or
half step. Along the orbifold boundary at the bottom
are all the
chords which have two identical pitches. These
four neighbors, except the unison chord, which has only two
The major and minor triads have five different triads as
six connections, counting the two ways they can lead into
The physical model shown is 5.5 inches long, made of nylon by SLS. The stl file is here. A high-res photo is here, with enough detail that you can look closely to see the tiny stairsteps of layered fabrication.
|Fractal polyhedra clusters
be derived by putting copies of one polyhedron at the
another. Scale the small polyhedra so they just touch
neighbors. Now take these clusters and position copies of
them at the
vertices of a larger polyhedron. Repeat to any depth level.
At right is an icosahedron of icosahedra of stellated dodecahedra. More info and the stl file to make your own are available here.
|Here is a set of
of geometric sculpture designs by Morton Bradley.
More information about these forms and Bradley's original work, and the stl files to make your own copies of these, are available here.
|Here are some solids of
consatant width. They generalize the Reuleaux triangle to
3D. If you
put one between two parallel planes, its width is 1 inch, no
what direction you measure. If they are resting on a flat
can put a sheet of glass on top of them and it slides around
staying perfectly level. An infinite number of such
possible and I just chose three different shapes that I
You can make your own with these stl files: 1, 2, 3.
|This first is an "elevated icosidodecahedron"
composed of 120 open equilateral triangles.
The model shown in my hand at right is made on an FDM machine. I have also made a wooden model of this and other Leonardo constructions, shown here.
The STL file is available here (0.43 MB) for you to produce on your RP machine.
|This second is a spherical form with 72 open
(24 triangles and 48 trapezoids). The form was
teaching model to illustrate one of the constructions in
(Book 12, proposition 7).
The model shown is about 3 inches in diameter, made on a Zcorp machine.
The STL file is available here (0.19 MB) for you to make your own.
|Here is a model of a
from small units. The original is a sixteenth century wooden
on display in a museum in Germany. It has seventy-six
units linked into a cycic chain. Whether it was functional
or a puzzle or just a curiosity is not clear.
This is a reconstruction that you can build on a rapid prototyping machine to help you think about how and why it was made. For more information about this curious object and .stl files for you to make your own copy, see this web page.
|Here is a 9-inch diameter sculpture made on a
color Zprinter by 3Dsystems. It is based on a stellation of
the icosahedron, with openings I added to make it more
interesting, and colored with a simple algorithm that helps
bring out its structure. To reproduce it in color, you
need a color Zprinter and a 3D file format that describes
colors, such as this.
Or if you have a standard 3D printer, you can make a
monochrome version from this
There are a couple of dozen more color Zprinter works on this page.
|This is six nested truncated icosidodecahedra
open structure. Be sure to notice the conical
connect the different layers together.
The model shown is about three inches in diameter, made in SLA. You can download the STL file (0.47 MB) if you want to make your own.
A short paper which explains the structure is available here.
|Shown at right is an SLS model about 3 inches
of a ball made of approximate rhombi. It is a good
understanding spatial symmetry, as it has the rotational
symmetry of an
but no planes of mirror reflection. It is also something of
to the mathematician/astronomer/instrument maker Abraham Sharp.
You can download the STL file (0.99 MB) and make a copy on your own RP machine. Then search for the twelve points where five rhombi meet. (In the image one is at the very bottom, and two are on a horizontal line halfway up.) Notice that they do not point directly towards each other. The idea behind the geometry is described in this paper.
|Here is one I call Tangled Reindeer.
is an SLS model, about 3 inches in diameter. The idea
a pile of reindeer with their bodies at the center.
You can have
a copy of your own by building this
Here is a paper describing the methods underlying this and some other recent RP sculptures of mine.
|This is a woven assemblage of Salamanders, in
homage to M.C. Escher. It is a prototype for a large sculpture
in a group
barn raising I led when I was artist-in-residence
MIT. Shown at right
is a small 2.5 inch diameter model made on a new
333 machine. This stl file
(0.54 MB) is
available if you want to make your own copy.
Here is a paper describing the methods underlying this and some other recent RP sculptures of mine.
|This is a small SLS model of a Snarl,
which is both a puzzle and a sculpture. This 2-inch diameter
scale is a
bit small for seeing all the details of the design. To fully
it, you can
tape together a larger paper model of it, using the template
in this short paper that I
presented at the
G4G6 conference in March, 2004. To make your own rapid
of the solution use this stl file
|Here is a synthesized form reminiscent of a
"sand dollar" (stl
1.7MB) and another which is a kind of spiral toroid creature
(stl file, 6MB).
Both are from a
project I call Echinodermania I, II, and III. A
describing the generating
techniques is available here.
I have added some images of additional forms from that paper on a page of More Echinodermania.
|This sculptural form is based on the (10, 3)-a
network, described in A.F. Wells' 1956 book The Third Dimension in
Recently the same underlying crystal structure has been
Toshikazu Sunada, who called it the
in various directions are very different, so see more photos
fascinating 3.5 inch nylon model here.
The .stl file for it is here if you want to make your own physical copy with solid freeform fabrication equipment.
|And now for something
This is an organic-looking form produced by an algorithm for "growing" shapes as cellular structures. It is actually just one moment in a 3D "movie" which produces a series of related forms, each with one more cell than the previous form. You can understand the algorithm by watching it grow on this page, which also has a range of additional examples.
Here is the stl file (2.6MB) for this particular moment in its development. As shown at right, it was made in ABS by FDM, with a height of six inches. The white material and the small size of the photo makes it hard to see the cellular structure. This larger photo of the model after I painted it makes very clear how it is constructed of many individual cells.
|Here is a puzzle made of twenty identical
that are fabricated on an FDM machine. More
file is available here.
|The algorithm described in the paper listed
be of interest to people who wish to convert sets of line
models for RP production. The method it describes for
segments with triangles is one of the steps in several of
above. At right is a 5 inch diameter form with five
produced by this method.
George W. Hart, "Solid-Segment Sculptures," presented at Colloquium on Math and Arts, Maubeuge, France, 20-22 Sept. 2000, and published in Mathematics and Art, Claude Brute ed., Springer-Verlag, 2002.
I later improved the algorithm to be more robust and have more adjustable parameters. A paper describing the later version is here:
G. Hart, "Sculptural Forms from Hyperbolic Tessellations," Proceedings of IEEE Shape Modeling International 2008, pp. 155-161 (online pdf version)