Some Solid (Three-dimensional) Geometrical Facts about the Golden Section
Having looked at the flat geometry (two dimensional) of the number Phi,
we now find Phi in the most symmetrical of the
three-dimensional solids - the Platonic Solids.
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The five regular solids (where "regular" means all sides are equal and all angles are
the same and all the faces are identical) are called the five Platonic solids after
the Greek philosopher and mathematician, Plato. Euclid also wrote about them.
For more information on these two famous Greeks, see the footnote.
Dice shapes
What shapes make the best dice?
We need to make sure all the faces are the same shape and that all the angles
and sides are equal, or some faces will be favoured more than others and so our dice will
be "unfair".
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
Coordinates and other statistics of the 5 Platonic Solids
They are the tetrahedron, cube (or hexahedron),
octahedron, dodecahedron and icosahedron.
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 Tetrahedron
Solid View
Stereo View
Wireframe Views
4 vertices with coordinates:
(1, 1, 1), (1, -1, -1), (-1, 1, -1), (-1, -1, 1)
6 edges of length 22,
mid point of edge to centre of solid = 1
4 triangular faces,
Surface area = 8 √3, volume = 8/3,
The Cube or Hexahedron
Solid View
Stereo View
Wireframe Views
8 vertices with coordinates (±1, ±1, ±1), i.e.:
(1, 1, 1), (1, 1, -1), (1, -1, 1), (1, -1, -1),
(-1, 1, 1), (-1, 1, -1), (-1, -1, 1), (-1, -1, -1)
12 sides of length 2, mid point of edge to centre of solid = √2
6 square faces each of area 4, surface area = 24, volume 8
The Octahedron
Solid View
Stereo View
Wireframe Views
6 vertices with coordinates (±1, 0 0), (0, ±1, 0), (0, 0, ±1) i.e.:
(1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0), (0, 0, 1), (0, 0, -1)
12 sides of length 2,
mid point of edge to centre of solid = 1/2
8 triangular faces,
surface area = 4√3, volume 4/3
There are two more important relationships between the dodecahedron and the icosahedron.
First, the mid-points of the faces of the dodecahedron define the points
on an icosahedron and the mid-points of the faces of an icosahedron define a dodecahedron.
The same is true of the cube and the octahedron. If we try it with a tetrahedron, we just
get another tetrahedron. Each is called the dual of the other solid where
the number of edges in each pair is the same,
but the number of faces of one is the number of
points of the other, and vice-versa.
Golden sections in the Dodecahedron,
Icosahedron and Octahedron
If we join mid-points of the dodecahedron's faces,
we can get three rectangles all at right angles to each other. What's more, they are
Golden Rectangles since their edges are in the ratio 1 to Phi.
The same happens if we join the vertices of the icosahedron since it is
the dual of the dodecahedron.
Using these golden rectangles it is easy to see that
the coordinates of the icosahedron are as given above since they are:
(0,± 1, ± Phi), (± Phi, 0, ± 1), (± 1, ± Phi, 0) .
You do the maths...
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.]
A Cube in a Dodecahedron
We can see a cube in a dodecahedron if we use one diagonal on each face. Since
the diagonals of a dodecahedron are Phi times as long as the sides (see
Pentagons and Pentagrams on the Phi and 2D Geometry
page at this site), then cube's
sides and the dodecahedron's sides are in the golden ratio.
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.
An Icosahedron in an Octahedron
Using the same three golden rectangles at right-angles to each other, we can also make
an octahedron.
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 Greeks, Kepler and the Five Elements
The Greeks saw great significance in the existence of just 5 Platonic solids and they
related them to the 4 ELEMENTS (fire, earth, air and water) that they thought
everything was made from. Together with the UNIVERSE, they associated each with a particular solid.
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, the tetrahedron has the smallest volume for its surface area
and the icosahedron the largest; they therefore
show the properties of dryness and wetness respectively
and so correspond to FIRE and WATER.
The cube, standing firmly on its base, corresponds to the
stable EARTH but the octahedron which rotates freely when held
by two opposite vertices, corresponds to the mobile AIR.
The dodecahedron corresponds 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.
Kepler called the golden section
"the division of a line into extreme and mean ratio", as did the Greeks.
He wrote the following about it:
"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)
Raoul Martens recommends an article in German on Kepler's
interest in the Platonic solids:
Die kosmische Funktion des Goldenen Schnitts by Theodor Landscheidt in Sterne, Mond, Kometen, Bremen
und die Astronomie zum 75. Jahrestag der Olbers-Gesell-schaft Bremen e.V.
Verlag H. M. Hauschild, Bremen 1995.
On the Flat Phi page, we saw that
the two triangles that
appear in the pentagram and pentagon were used by Roger Penrose to design
tiling patterns with five-fold symmetry
called Penronse tilings. Is there a three-dimensional analogue of those two-dimensional
tilings? The answer, thought to be impossible until Penrose's work of the early 1970's
showed that there could be structures that filled space (in the same way that tilings fill planes)
that have five-fold symmetry.
Are any Platonic solids space-filling?
The Cube
Yes, since identical copies of a cube can be stacked to fill a volume of space
as large as we like with no gaps. Any child playing with cubical bricks
learns how to build a wall or a bigger cube from the little ones, and the process
can be continued indefinitely to fill all space.
Tetrahedrons and Octahedrons
The tetrahedron and the octahedron
used together will fill space with no gaps
since one fills the gaps left by the other.
But neither the
tetrahedron on its own nor the octahedron on its own will
pack space without leaving gaps.
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.]
Icosahedrons, Dodecahedrons
Similarly, neither will the icosahedron nor the dodecahedron,
as it is analogous to trying to tile a plane with pentagons - they leave odd gaps that
are not pentagonal and both the dodecahedron and the icosahedron exhibit five-fold
symmetry too. To see this, look back at the sections above on the Icosahedron and
Dodecahedron and you will find that, in the "other views" each has a view with five-fold
symmetry. These views correspond to looking along an axis through the centre of the
solids which have five-fold symmetry.
Quasicrystals
Penrose found that there are two simple shapes that you can use to
fill a flat space as large as you like and that have five-fold axes of symmetry.
The shapes are built from 6 flat faces which are rhombuses, that is, shapes
with all sides of equal length (like a square) and which has opposite sides
parallel (again like a square), but which does not have all its angles equal - so they
are diamond shaped (rhombs, rhombuses). The Penrose tiling shown on the Flat Phi page
is also made from two rhombuses and fills the plane with a five-fold symmetric pattern.
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
the ratio of the two diagonals of the diamonds (rhombuses) is Phi!
So this is a different rhomb from the Penrose rhombs and
we shall call it the golden rhomb.
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^{r}
and 2x58·282525588° = 2x1·0172219674^{r}.
[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.
Do quasicrystals occur in nature too?
Yes they do and a large number of substances have now been identified with such structures.
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
symmetryPhysics Review Letters 1984, Vol 53, pages 1951-1953.
References and Links
H S M Coxeter's
Regular Polytopes,
(Third Ed) 1973,
Dover, is a very popular book at an amazingly low price - well worth getting!
H S M Coxeter,
Introduction to Geometry, 1961,
John Wiley, is a classic! See especially section 11.2: De Divina Proportione.
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 we use three connected squares we find
2 different triomino shapes can be made.
If you have played Tetris then you will know
that there are 5 shapes of 4 square tiles (tetrominoes) as shown here.
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 hexominoes
(6 squares) can you find? What about heptominoes (7 squares)?
[Answers]
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.
Footnotes
The Greeks from
Euclid ( around 300BC)
and before knew that there were only 5 solid shapes with all sides
equal, all angles equal, so that the faces are regular polygons.
Plato
They were also mentioned by the Greek philosopher
Plato (428BC-348BC).
He established an Academy in Greece and the motto over the entrance was
Let no one ignorant of geometry enter here
As a philosopher, he held the view that mathematical objects "really" existed
so that they are discovered by mathematicians (in the same way that new
continents are discovered by explorers) rather than invented
in the way that the TV or computer were invented. Plato believed that mathematics
provided the best training for thinking about science and philosophy.
The five regular solids are named "Platonic Solids" today after Plato.
Euclid
The most famous ancient book on geometry was written by
Euclid (pronounced "U - klid") who lived around 300 BC and worked
at the Library at Alexandria in Egypt, the foremost centre of learning in the world
at that time.
Actually, the book was a collection of 13 volumes, called The Elements and was
the collected knowledge on geometry, superbly arranged and logically presented.
It was the standard mathematics text book in Europe for centuries
because it trained the reader to think logically, only relying on results
that could be proved logically from self-evident starting points (axioms).
Here are some axioms:
Things that are equal to the same thing are equal to each other.
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.
From these, Euclid proved theorems such as
The angles in a triangle add up to two right angles.
One of Euclid's aims in his Elements seems to be
to prove that there were only 5 solid (i.e. 3-dimensional) objects with all sides
equal and all angles equal, and this occupies the final (13th) book of the Elements.
The 13 books - now available in English in a 3-volume set -
are classics is every sense!
We saw above that the Greeks knew of the 5 shapes that make fair dice.
The Romans used a cubic dice and this is the one we most often use today.
Should we say one die or one dice?
The dictionary says that die is singular and dice is its plural form,
so we ought to speak of throwing a die or two dice.
These days the plural word dice is 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 called casting the dice. Some of the Roman soldiers "cast lots" for
the clothes of Jesus at his crucifiction. Today we still use the
phrase the 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 being
cast in the heroic mould as 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 word die.
The real meaning of the phrase the die is cast is
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 make dice refer to the singular as well as the plural.
From the Platonic solids that we saw above, we have dice of
4 sides : the tetrahedron
6 sides: the cube (or hexahedron)
8 sides: the octahedron
12 sides: the dodecahedron
20 sides: the icosahedron
There 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 fair dice - i.e. each number on a face
is as likely as any other number to turn up.
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.
Putting both of the above shapes together, we get a dice which is two n-gon-al
pyramids, joined at their bases (the n-gons) to form a double pyramid or
bi-pyramid.
The picture shows a 12-sided dice formed
from two 6-sided pyramids joined at their hexagonal bases.
Perhaps we should call it a bi-hexahedral dice.
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.
Iso-hedral shapes
Here is Ed Pegg Jr.'s
complete list of ALL the 3-D dice shapes
which have every face identical.
It includes all our 5 Platonic solids, and,
since it also includes those where not every edge is the same length,
it includes the bi-pyramids too. Every face is identical to
every other face, so all the faces have exactly the same polygonal shape,
but some edges have different lengths to others. There are others apart from
the Platonic solids and the bi-pyramids and are some pretty weird too!
The common feature is that all of them would make good dice.
Since every face is the same, they are called iso-hedral.
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 Game shops or online, for example at
Crystal Caste .