- Phi and 3-dimensional geometry

From 2-dimensional (flat) shapes, we turn to 3-dimensional ones (solids).- Dice Shapes

We need symmetry in dice if they are to be fair, but is the cube the only possible shape? No, there are 5 and only 5 fair dice shapes: - Coordinates and other statistics of the 5 Platonic Solids

Some other relationships between these shapes... - The Greeks, Kepler and the Five Elements

- Dice Shapes
- Quasicrystals and Phi
- References and Links
- Two Footnotes

1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

The dice you usually find today are cube-shaped - 6 square faces, all angles are right-angles and all sides are the same length.

[There *are* other shapes that make fair
dice if we relax these conditions a little. Can you guess what they are?
See the footnote for the answers.]

There are only FIVE fair-dice-shapes altogether
if we strictly insist on the following conditions:

all sides are equal in length and

all angles are equal so that

all the faces are identical in shape and size

all angles are equal so that

all the faces are identical in shape and size

Their names come from the number of faces (hedron=face in Greek and its plural is hedra). tetra=4, hexa=6, octa=8, dodeca=12 and icosa=20.

Remember that in these pages Phi is 1·61803.. and phi is 1/Phi = Phi-1 = 0·61803... .

The **Solid** images can be rotated ( press the button) as can the **Stereo views**.
For the auto-stereographic views, either cross your eyes or keep your eyes focussed in the distance until the
two images fuse into one and you see the shape in depth. If you
place your mouse on the "rotate" button before
you do this then a quick click will make it appear to rotate in 3-dimensions.

You may notice a pause when first seeing a rotating diagram while it is being
downloaded.

The "wire-frame" views are symmetrical plan views of the frame of the object with wire edges and the faces missing, dotted lines being edges that would be hidden by solid faces.

The same happens if we join the vertices of the icosahedron since it is the dual of the dodecahedron.

Using these

(0,± 1, ± Phi), (± Phi, 0, ± 1), (± 1, ± Phi, 0) .

- Here is an interesting way to make a
**model of an icosahedron**based on the three golden rectangles intersecting as in the picture above:- Cut out three golden rectangles. One way to do this is to take three postcards or other thin card and cut them so they are 10cm by 16.2cm.
- In the centre of each, make a cut parallel to the longest side
which is as long as the shortest side of a card.

The three cards will be slotted through these slits to make the arrangement in the picture above. To do this, on one of the cards extend the cut to one of the edges.+--------------+ Make and one +-------------+ ! ! two of ! ! ! ====== ! of these ! =========== ! ! these ! ! +--------------+ +-------------+

- Assemble the cards so that they look like the picture here of the red, green and blue rectangles. [This is a nice little puzzle itself!] You may want to put pieces of sticky-tape where two cards meet just to make it a bit more stable.
- Now you can make an icosahedron by joining the corners of the rectangles by glueing cotton so that it looks like the picture above.
- It will be quite delicate, so tape another piece of cotton to one of the short edges of one of the cards and hang it up like a mobile!

- If you are good at coordinate geometry or like a challenge, then show that the
12 points of the icosahedron divide the edges of the octahedron in the ratio Phi:1 (or
1:phi if you like)
where the octahedron has vertices at:

( ±Phi^{2}, 0 0 ), ( 0, ±Phi^{2}, 0 ), ( 0, 0, ±Phi^{2})

[from H S M Coxeter's book**Introduction to Geometry**, 1961, page 163.]

With thanks to Prof Susan Goldstine.

In fact, 5 distinct cubes can be fitted into the dodecahedron, with the vertices of the cubes meeting at the vertices of the dodecahedron as shown in the pictures below.

Notice that:

- The cubes are red, green, yellow, blue and magenta.
- Each of the 5 cubes has 12 edges, totalling 60 in the dodecahedron and so each of the 12
faces of the dodecahedron will have 5 of the edges in it.

In fact, each face of the dodecahedron will have a**pentagram**on it formed from*just one of the edges of each of the 5 cubes*. - No edges will overlap in any of the cubes but each cube edge will cross the one edge of 2 of the other cubes.
- Each of the 5 cubes has 8 vertices making 40 cube-corners to share among the 20 vertices of the dodecahedron. Each vertex of the dodecahedron is shared with exactly 2 of the 5 cubes.

If we put a square as shown around each rectangle, the squares will also be at right angles to each other and form the edges of an octahedron.

Now if we join the "golden-section points" forming the corners of our three rectangles (and now on both the edges of an octahedron and also forming the vertices of an icosahedron as we saw above), we can see how to fit an icosahedron into an octahedron - and the process involves golden sections!

Here are some more Platonic-solids-within-Platonic-solids:

** A Tetrahedron in a Cube**

Select one corner of a cube and join it
to the *opposite* corner on each face.

** An Octahedron in a Tetrahedron**

Join the mid-point of each edge to any other
edge mid-point where the connecting line lies on one face of the tetrahedron.

** An Octahedron in a Cube**

Join the mid-points of faces: if two faces are
next to each other at a corner, then their mid-points can be joined.

The astronomer and mathematician, Kepler (1571-1630), shown here as a link to the History of Mathematics web site at St Andrews University, Scotland, justified this as follows:

Of the 5 solids, theKepler called the golden section "the division of a line into extreme and mean ratio", as did the Greeks. He wrote the following about it:tetrahedronhas the smallest volume for its surface area and theicosahedronthe largest; they therefore show the properties ofdrynessandwetnessrespectively and so correspond to FIRE and WATER.

Thecube, standing firmly on its base, corresponds to the stable EARTH but theoctahedronwhich rotates freely when held by two opposite vertices, corresponds to the mobile AIR.

Thedodecahedroncorresponds to the UNIVERSE because the zodiac has 12 signs (the constellations of stars that the sun passes through in the course of one year) corresponding to the 12 faces of the dodecahedron.

Raoul Martens recommends an article in German on Kepler's interest in the Platonic solids:"Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel."

Johannes Kepler, (1571-1630)

1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

To see this, look at the "Tetrahedron In A Cube" picture above. We can imagine a kind of 3-D "graph paper" filled with cubes and in one cube we will place a tetrahedron. It uses just 4 of the 8 vertices of its cube, so at each of these vertices in all the cubes at those vertices, we place more tetrahedra. We repeat on the surrounding cubes and in so doing, we split all the vertices into two sets - those on a tetrahedron and those not on one. Each vertex not on a tetrahedron is the centre of one octahedron. To see this visit Mark Somer's web pages of Amy C. Edmondson's book A Fuller Explanation about the geometry of R Buckminster Fuller, chapter 12 Figure 12.2 for a helpful diagram. [The online book is worth browsing through as it has lots more interesting geometry about space filling shapes.]

For the solid shapes, the faces are all diamonds (rhombs) but not the ones used in the Penrose tiling and pentagons and pentagrams. The surprising relationship that holds for these new rhombuses is that

This makes the semi-angles (half the angles inside the rhombus) have tangents of Phi and phi so the angles of the rhombus are 2x31·717474..° = 2x0·55357435889

[The angles in the rhombs in the Penrose tiling are 2/5 pi and 3/5 pi (72° and 108°) in one and 1/5 pi and 4/5 pi (36° and 144°) in the other.]

The two solids are similar to a cube but the faces are golden rhombs.
The first shape is made by attaching three golden rhombs at their shorter
angles in the same way as three squares meet at
a corner of a cube. A duplicate is made and the two fit
together to make a six-sided shape like a slanted cube.
This is called a **prolate rhombohedron**.

The other shape is made by joining three golden rhombs together in the same way but
at the larger angles this time. A duplicate of this is again fitted to make a different
six-sided cube-like shape. This is called an **oblate rhombohedron**.

The two shapes look like cubes leaning over to one side.

Take a large number of one of these shapes and you can indeed fill as large a
space as you like with them.
When stacking cubes or octahedra, all the shapes are aligned
identically (look identical, with the same orientation).
When we use a rhombohedron, some must be turned round to fit in with others.
These also occur in nature, although only discovered since the 1950's
and, because they are not quite as symmetrical as crystals, as called
**quasi-crystals**.

Crystals, the most symmetrical structures (with identical orientation for all the building
blocks) are seen in sugar and salt as well as diamonds and quartz.
Quasicrystals are an unsuspected new state of matter, sharing some of the properties
of crystals and also on non-crystalline matter (such as glass). In 1984 the "impossible"
five-fold symmetry was observed in an aluminium-manganese alloy (Al_{6}Mn)
and the term
quasicrystal was invented for it in:

D Shechtman, I Blech, D Gratias, J W Cahn **Metallic
phase with long-range orientational order and no translational
symmetry** *Physics Review Letters* 1984, Vol 53, pages 1951-1953.

H S M Coxeter, Introduction to Geometry, 1961, John Wiley, is a classic! See especially section 11.2:

The classic and encyclopaedic book on tilings is Grunbaum and Shepard's Tilings and Patterns W H Freeman and Co, 1989. It is well worth dipping into just to admire the pictures and patterns as the maths in it is sometimes beyond school level i.e. post age 16. Nevertheless, it is an inspiring book and chock full of interesting results on all kinds of patterns and tilings with copious illustrations. There is also a cheaper paperback version but both forms are now out of print. If you can pick up a second-hand copy, it's well worth the effort if you want to study tilings seriously.

Polyominoes: A Guide to Puzzles and Problems in tiling George Martin, the Mathematical Association of America (1991), 184 pages, is about those wonderful polyominoes puzzles. A polyomino is an extension of a domino of which is just 1 shape that we can make from two squares.

Domino | 2 Triominoes | 5 Tetrominoes | 12 Pentominoes |

If you have played Tetris then you will know that there are 5 shapes of 4 square tiles (

A puzzle I came across when I was 12 is a box of the 12 pentominoes, each piece being one of the ways of "tearing 5 connected square stamps from a perforated sheet" as George Martin puts it. They fit into a variety of shaped boxes but mine came in a 10x6 box. They can be fitted into it in 2339 different ways (ignoring rotations and horizontal and vertical flips of the box).

Puzzle Can you fit all 12 pentominoes in a box of size 12x5? 15x4? What about 3x20? [Answers].

So from domino, triomino (or triomino), tetromino, pentomino, etc we get the family of Poly-ominoes!

Puzzle How many

Each set has its own mathematical jigsaw puzzles, but, to me, they are far more interesting than the pictorial jigsaw puzzles in games shops. I must make a separate page on these sometime.

Andrew Clarke's Poly Pages have a profusion of pleasing polyomino patterns and puzzles to ponder and loads of luring links too.

A definite "must buy" is Polyominoes by Solomon Golomb and Warren Lushbaugh, a Princeton University press paperback (1996) of 198 pages. Golomb is the inventor of polyominoes and this is the revised and expanded second edition of the original of 1965 that sparked off the polyomino puzzle craze.

Fractals, Chaos and Power Laws, M Schroeder, W H Freeman publishers, 1991. This is another fascinating book with much on self-similar sequences and patterns, Fibonacci and Phi. I have found myself dipping into this book time and time again. There is a chapter on the forbidden five-fold symmetry and its relation to the Fibonacci rabbits. (More information and you can order it online via the title-link.)

Robert Conroy has a page with lots of wire-frame pictures of other three-dimensional structures that are related to the Icosahedron and Dodecahedron.

If your browser has a VRML plug-in, then check out George Hart's Virtual Polyhedra site with over 700 polyhedra to manipulate on-screen!

Mark White's Geometry pages have some wonderful illustrations and more about combinations of the five Platonic solids.

Let no one ignorant of geometry enter hereAs a philosopher, he held the view that mathematical objects "really" existed so that they are

Here are some

Things that are equal to the same thing are equal to each other.From these, Euclid proved

The whole is greater than the part.

It is possible to draw a circle with any point as centre and with any radius.

It is possible to draw a straight line between any two points.

The angles in a triangle add up to two right angles.One of Euclid's aims in his

The 13 books - now available in English in a 3-volume set -
are classics is every sense!

This 3-volume set is
cheap and has been the standard version for many years:

The Thirteen Books of Euclid's Elements, Books 1 and 2

The Thirteen Books of Euclid's Elements, Books 3 to 9

The Thirteen Books of Euclid's Elements, Books 10 to 13

all are by Thomas L Heath who translated and annotated
Euclid's work, each is about 464 pages, published by
Dover in paperback, second edition 1956.

Should we sayone dieorone dice?

The dictionary says that die is singular and dice is its plural form, so we ought to speak ofthrowing a die or two dice.

These days the plural worddiceis often used for one die and the dictionary recognises this also.

A popular gambling game from at least Roman times involved throwing dice and is also calledcasting the dice. Some of the Roman soldiers "cast lots" for the clothes of Jesus at his crucifiction. Today we still use the phrasethe die is cast. I used to think this phrase meant that a mould (US spelling=mold) had been made since we also read of someone beingcast in the heroic mouldas if they had been molten metal poured into a mould from which they solidify into a heroic shape. However I was wrong and it is just another use of the worddie.

The real meaning of the phrasethe die is castis that a dice (one!) has been thrown (cast) meaning that, as in a game of chance, "the outcome is now fixed, the decision is made".

In these pages, I shall stick to the popular and common use, and makedicerefer to the singular as well as the plural.

From the Platonic solids that we saw above, we have dice of

4 sides : the tetrahedronThere are other shapes if we don't insist that all the sides are the same length OR we allow 2-D shapes, but which still are

6 sides: the cube (or hexahedron)

8 sides: the octahedron

12 sides: the dodecahedron

20 sides: the icosahedron

If we let sides be different lengths, we can have a ** prism** which is like a
new (unsharpened) pencil with flat sides.
Often pencils have just 6 flat sides, and we roll the pencil so that any side
is likely to be face up. We can imagine a pencil with 8 sides, or 7 or even 27.
If we have an odd number of sides, no one face is "up" (consider a triangular cross-sectioned
pencil for instance, with just 3 choices of side). Here we may agree to use the side that
the pencil lands on.

The other range of shapes is the **spinner** that comes with some boxed games.
Here we have a flat polygon with all sides of the same length (to make it fair). This was
not in our list of Platonic solids because it is not a *solid* - it is just a flat 2-dimensional
shape.
However, we *can* have any number of sides and each is equally likely to be
the side the spinner lands on, so it is fair.

If we used pentagons then the bi-pyramidal dice would be 10-sided.
It would be useful for generating random numbers up to 10.

By using two of them,
say a red one for **tens digits** and a
green one for **units digits**,
we can roll random numbers from the hundred values between
00 and
99.
If we added a blue one also,
then we can get up to
999, and so on.

The advantage of the bi-pyramidal dice is that
*there is always a side on top*
no matter how the dice lands.

It includes all our 5

The common feature is that all of them would make good dice.

Since every face is the same, they are called

With thanks to Robert Popa for recommending the following:

More isohedral (i.e same-faced) dice
where the edge lengths are not equal but the faces are all identical.

You can purchase some of these non-standard dice at
Crystal Caste or
Plane of Games.

[Back to the main text.]

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© 1996-2006 Dr Ron Knott

updated 26 May 2006