ITSPHUN geometry

Each ITSPHUN construct is a model, or approximation, of some abstract, mathematical object.

The relation between the octahedron and an ITSPHUN model (built with our spiral-shaped wood pieces) is illustrated below.

The relation between the octahedron and an ITSPHUN model (built with our spiral-shaped wood pieces) is illustrated below.

ITSPHUN "octahedron"

Spiral piece and its underlying polygon

ITSPHUN activities

Here are some fun activities, games and puzzles you can do with ITSPHUN pieces.
__Contact us__ if you have any suggestions,

Build objects from a model or picture

This is a great way to develop spatial visualization and pattern recognition skills. Check out our
__photo gallery__ or download the file at right to get you started.

For an extra challenge, try counting how many pieces are required before starting to build your model.

For an extra challenge, try counting how many pieces are required before starting to build your model.

Have a ball!

- Build ball-like objects like the
__dodecahedron__,__rhombicuboctahedron__,__truncated icosahedron__(soccer ball), etc. With more pieces you can build bigger and bigger ball models, right?

Rhombicosidodecahedron

Billions and billions of stars

- Build some star-like models like the
__stellated octahedron__(stella octangula).

Stella octangula

Build by connecting

Two objects that contain a same-shaped piece can be joined by removing that piece from each and then connecting the freed notches. The picture shows two
__dodecahedra __
(red and yellow) and a
__pentagonal prism__
(blue). They are connected by common pentagonal faces.

What cannot be built with ITSPHUN?

- Try to make a Möbius strip with ITSPHUN pieces.

Build by connecting, part 2

The model at right can be build as follows:

- Build a
__dodecahedron__ - Connect, as described above, 12
__pentagonal pyramids__to the 12 faces of the dodecahedron - Connect a
__tetrahedron__to each triangular face of each pentagonal pyramid

Build from description

- Build a polyhedron that has only square faces and hexagonal faces and has the following properties: (a) each square has only hexagonal faces as neighbors, and (b) each hexagonal face has three square and three hexagonal neighbors that alternate.

Build with colors

- For a given object, what is the minimum number of colors needed so that no two same-colored faces touch each other along an edge?

*Which objects can be colored in this manner with just two colors?*

- What if two faces of the same color are not allowed to touch even at a vertex?

- For a given object, how many faces of the same color can you place so that they do not touch each other at an edge/corner?

Cuboctahedron:

same color faces do not touch - two colors are sufficient

Build as a tag game

- Build an object as follows. Choose the first piece (the it piece). You can only make connections to the it piece. After each connection, the piece that is connected to itbecomes the new it (this can be a new piece or one that is already part of the object). Continue until you cannot make any new connections ("it" is already connected on all sides). Can you build the entire object (make all connections) in this manner?

For example, in the picture at right, assume Red is it. After connecting it to Green,

Green becomes it. After the connection Green-Yellow, Yellow becomes it, and then Blue.

If we connect Blue to Green, Green is it again and we need to add a new piece to Green, etc.

Does it matter:

- which piece you choose as your starting it, and
- in which order you make the connections?

- Can you predict beforehand which objects can be completed this way?

- Change the game so that a new it can only be a new piece that you're adding (make all connections to the old it before moving to the new one). Which objects can you build now?

Platonic solids - yes, cuboctahedron, icosidodecahedron - no (prove it!), etc.

Fun with words

Increase your vocabulary! Impress your friends! Improve your diction!

- Build the
__Johnson solid__with the most impressive name. Casually mention the name to anybody who asks what you are doing (make sure to practice saying the name aloud).

- Build all Platonic/Archimedean/Johnson solids with fewer faces than letters in their names. Can you find any that have exactly as many faces as there are letters in their name?

In turn, a player picks a card and reads aloud the name of the corresponding Johnson solid. If (s)he stumbles then (s)he must keep the card, otherwise (s)he discards it. After 5 rounds each player must build the objects on the cards (s)he has. First one to finish wins, last one must clean the room, or read aloud the list of all 92 Johnson solids (while standing on one leg), or just buy the next round of drinks.

Gyroelongated pentagonal cupolarotunda

Geometric art

- Build a
__bilunabirotunda__ - Build a
__pentagonal orthobicupola__ - Connect them to make the head and body of the owl; add the beak, legs, and tail. (Does an owl even have a tail?)

- Build your own creations

Geometric art game

Each team builds a sculpture; the most creative/decorative/symmetric/funny/colorful/tall/etc. sculpture wins. Taking turns, each team member must add 3 pieces to the growing sculpture.

- No communication is allowed between team members.
- Limited, verbal only, communication is allowed (no pointing).
- Removing pieces is also allowed - each team member is allowed 3 "moves", a move being either adding or removing a piece from the sculpture.

Near misses and large balls

Build some
__Goldberg polyhedra__
using hexagons and exactly 12 pentagons. We said above that you cannot build larger and larger convex polyhedra - what's the catch?

Goldeberg polyhedron G(2,1)

Build with no rules

Bend and twist the ITSPHUN pieces to connect them in ways they were not meant to be connected!

- Build the model at right

Build to infinity

Certain polyhedra can be stacked together, in all directions, without leaving any holes or gaps. A familiar example is the cube; a less familiar one is the truncated octahedron. We can join (see "build by connection" above) such space filling objects forever and ever, creating larger and larger structures, and using more and more ITSPHUN pieces.

- Connect a few
__truncated octahedra__to get a feel on how they stack together to fill the space. - If we continue adding new such space-filling polyhedra, in all possible ways, ad infinitum we'll end up with an infinitely large object containing infinitely many pieces, right?

Another way to think about it: every time we join two objects we remove the "interior" common face(s) and leave only the exterior faces. The partial constructs contain only these exterior faces that form the border of an increasingly large object. But, at infinity, the final, space-filling "object" has no border at all, and thus has no faces.

Build to infinity, part 2

Do not get discouraged by the previous activity - infinite polyhedra do exist! As a first example, start with the same space-filling with truncated octahedra.

Using 8 hexagons build a (partial) truncated octahedron (TO) without square faces.

Build a second TO with the opposite notch orientation and attach it to the first at a hexagonal face. Note that, unlike our usual connection method (see "build by connecting" above), we need to remove only one of the common faces when connecting - the internal face which is the intersection of the two TOs remains in the model.

Continue adding TOs until you fill the space (or until you run out of pieces).

Build a second TO with the opposite notch orientation and attach it to the first at a hexagonal face. Note that, unlike our usual connection method (see "build by connecting" above), we need to remove only one of the common faces when connecting - the internal face which is the intersection of the two TOs remains in the model.

Continue adding TOs until you fill the space (or until you run out of pieces).

Build to infinity, part 3

There are many infinite polyhedra.

- Build a model of the
__mucube__, another infinite regular polyhedron discovered by Coxeter.

- Find another infinite polyhedron related to the space filling with truncated octahedra (see part 2).

Counting

- Use 12 pentagons to build a dodecahedron. How many connections did you make? (or, how many edges are there in a dodecahedron?)

- How many vertices?

- Can we use the same method of counting edges for all (closed) polyhedra?

- Can we apply the same method of counting vertices to all polyhedra?

Beyond counting: interesting number patterns

- Build several objects, then count their number of faces (F), edges (E) and vertices (V). Verify
__Euler's formula__F - E + V = 2. - Build the Platonic solids and count their numbers of faces, edges and vertices. Can you find any patterns?

The tetrahedron is its own dual, and indeed F(tetrahedron) = V(tetrahedron).

Symmetry and color

- Build a cube using 6 squares of 3 colors, 2 squares of each color. Build several more cubes using squares of the same colors.

Compare the cubes: are they the same or different? What does "different" mean?

- How many different cubes have this color scheme?

- Does the answer change if we allow the cubes to be "turned inside out"?

(This can be done by swapping opposite faces.)

- Is one cube "more symmetric" than the others? In what way?

If you start with two faces of different colors then, after any rotation of the cube, the same faces will still have different colors.

- Repeat this activity with other coloring schemes (e.g. 2 colors, 3+3); note that symmetric cubes do not exist for all schemes.
- Repeat with other simple objects, e.g., octahedron.

Symmetry and color, part 2

- Build the 2 different cubes with the 3+3 coloring scheme (2 colors, 3 faces of each color).

Which cube rotations are compatible (as defined above) with each coloring scheme?

*the 120 and 240 degrees rotations around the cube diagonal from the blue vertex FUL to the orange vertex BDR (they preserve the colors exactly), and**180 rotations around the 3 semi-diagonal axes from one edge midpoint to the opposite edge midpoint: FR-BL, FD-BU, UR-DL (they swap the colors). Total: 6 rotations (including the "identity", i.e., the rotation that leaves the cube unchanged)*

*180 degree rotation around the face axis UD preserves the colors,**180 degrees rotations around the semi-diagonal axes FL-BR and FR-BL swap the colors. Total 4 rotations, including the identity.*

- Repeat the activity for other objects and coloring schemes.

Triangular and tetrahedral numbers

- Build the object at right (a tetrahedron-shaped partial model of the
__mutetrahedron__, another infinite polyhedron) using 40 hexagons of 4 colors. Note that the parallel planes have the same color (two hexagons have the same color if and only if they are parallel).

- How many green hexagons are needed to add another layer to this model?

- How many green hexagons are needed for the n-th layer?

- What is the total number of green hexagons in a tetrahedron with n layers?

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