DIY · Card 59 · Mathematics · Geometry
Six strips of card, ten pencil marks on each and some patience. Weave them together and they close into a light, springy ball with 12 pentagon holes and 20 triangles: a polyhedron you made from circles.
01The brief
Weave a perfect ball from six flat strips.
6
strips, each one a great circle round the ball
30
crossings: every strip meets every other strip twice
12 + 20
holes: twelve pentagons and twenty triangles
Six flat strips of card don’t look like a ball. But bend each into a ring, cross the rings at just the right points, and weave them over and under, and they pull each other into a near-perfect sphere.
It is the same weave as the rattan balls of sepak takraw, and its holes follow a shape mathematicians call the icosidodecahedron. With the planner below, you will know exactly where every crossing goes. It belongs with the geodesic dome, another way to make a sphere from straight-ish pieces, and the hyperbolic slot, curves built from straight lines.
02Kit
Six strips of card and a few paper clips. Six colours help a lot.
Tick off what you have
0 of 4 ready
03Method
Mark the letters first; then the weaving is just a matter of matching them.
Step 1 of 8 · Minute 0
You need six strips of thin card or strong paper, each 1.5 cm wide and 41 cm long, ideally in six colours, a ruler, a pencil, a glue stick or tape, and a handful of paper clips.
The ball is lopsided.
Why
The marks aren’t evenly spaced, or a strip is a different length.
Fix
Measure every strip and mark carefully. All six loops must be exactly the same size.
The last strip won’t fit.
Why
A crossing earlier on was matched to the wrong letter.
Fix
Check each crossing’s letters: strip A’s C-mark must sit on strip C’s A-mark. Undo and redo the wrong one.
The strips slide apart.
Why
Too few over-under crossings, or thin, slippery paper.
Fix
Weave over and under wherever you can, use thin card, and add a dab of glue under each crossing.
04The geometry
A great circle is the biggest circle you can draw on a ball, like the equator. Any two great circles cross at two points exactly opposite each other. Six of them can be arranged so evenly that all 30 of their crossings are spaced alike: on each strip, a crossing every 36°, ten in all.
Turn the ball and follow one strip round, then add strips one at a time:
Crossings
30
Pentagons
12
Triangles
20
V − E + F
30 − 60 + 32 = 2
Every strip is a great circle, the biggest circle you can draw on a ball. Any two great circles cross twice, on opposite sides, so 6 strips make 30 crossings. With all six there are 30, and the holes between them are exactly 12 pentagons and 20 triangles: the pattern of a shape called the icosidodecahedron.
Ready to build? Choose a size and get strips to cut, with every mark labelled:
Cut each strip
40.7 cm
Mark every
3.77 cm
Strip width
about 1.5 cm
Each strip meets the other five once, then the same five again in the same order, on the far side of the ball. Where strip A has a mark labelled C, strip C must have a mark labelled A: lay those two marks on top of each other. The pale end is the overlap for gluing the strip into a loop.
05The big idea
Twelve pentagons
close a ball.
Triangles and hexagons lie flat. To bend a pattern round into a ball, you need something with fewer sides, and the magic number is twelve pentagons. A football has twelve; a buckyball has twelve; your woven ball has twelve.
That isn’t a coincidence. Euler’s formula, corners − edges + faces = 2, forces it: any closed pattern made of hexagons and pentagons, with three edges at each corner, has exactly twelve pentagons, however many hexagons there are.
06The math
1 · Crossings
2 × C(6, 2) = 30
Every pair of the 6 strips crosses twice.
2 · Strip length
ℓ = πd, marks every πd ⁄ 10
Ball diameter d; plus a little for the overlap.
3 · Euler
30 − 60 + 32 = 2
Corners, edges and faces of the weave.
Each strip meets the other 5 strips twice, so it has 10 crossings and 10 segments between them. Six strips give 60 segments, the edges. The holes are 12 pentagons and 20 triangles, 32 faces. Corners minus edges plus faces: 30 − 60 + 32 = 2.
For a 12 cm ball, each strip goes round π × 12 ≈ 37.7 cm, with marks every 3.77 cm, one every 36°. At each crossing two strips meet at about 63.4°, the angle whose tangent is exactly 2.
07Real life
From a kicking game in Southeast Asia to molecules and viruses.
Sepak takraw
In this Southeast Asian sport, players kick a ball over a net without using their hands. The ball was traditionally woven from rattan in this six-strip pattern, with wide strips so that 12 pentagonal holes remain; today most are moulded plastic in the same design.
Footballs
The classic black-and-white football, first used at the 1970 World Cup, is another polyhedron blown up into a ball: 12 black pentagons and 20 white hexagons. Like the woven ball, it needs exactly 12 pentagons to close.
Buckyballs
In 1985 chemists discovered a molecule of 60 carbon atoms shaped like a football, named buckminsterfullerene after the architect of geodesic domes. Researchers have even used woven takraw balls as models for it.
Virus shells
Many viruses protect their genes inside a shell built from identical protein pieces arranged with the symmetry of an icosahedron, the same symmetry as your woven ball. It is one of nature’s favourite ways to enclose space with repeated parts.
Kagome baskets
Japanese kagome weaving crosses three sets of bamboo strips to make a pattern of triangles and hexagons. Weavers bend the same pattern into round baskets by adding pentagons, exactly as the six-strip ball does.
Euler’s formula
In the 1750s Leonhard Euler noticed that for any solid without holes, corners − edges + faces = 2. The cube gives 8 − 12 + 6; the woven ball, 30 − 60 + 32. It is one of the founding results of topology.
12
For centuries, players across Southeast Asia have kept a woven rattan ball in the air with their feet, knees and heads. The official rules of sepak takraw still describe the ball by its weave: a single woven layer with twelve holes.
Those twelve holes are the twelve pentagons of your paper ball. Weavers found, by hand and by eye, the same arrangement of six great circles that mathematicians describe with symmetry and Euler’s formula.
08Is and isn’t
You can make a ball from any number of strips.
Some numbers work much better than others. Six great circles can be arranged so evenly that every crossing looks exactly like every other; most numbers of strips leave a lopsided ball.
The holes should all be the same shape.
Three great circles at right angles do leave eight identical triangles, like an octahedron, but the ball is very lumpy. Six circles make a much rounder ball, at the price of a mix of pentagons and triangles.
You need glue everywhere to make it hold.
Woven over and under, the strips lock each other in place. Traditional rattan balls are held together by the weave alone.
It’s just a craft, not maths.
The ball is a precise geometric object, the icosidodecahedron, and its 30 crossings, 60 edges and 32 holes obey Euler’s formula exactly.
09Go further
A
A bigger ball
Use the strip planner to make a 25 cm ball from wider card strips. Is it stiffer or floppier?
B
Count with colours
Colour the pentagon holes on one side with dots. How many pentagons touch each triangle? How many triangles touch each pentagon?
C
Wide strips
Make a ball from strips 4 cm wide. What happens to the triangle holes? That is how a takraw ball looks.
D
Four strips
Try weaving just four great circles, crossing evenly. What shapes are the holes now?
E
A football
Draw a football’s pattern of 12 pentagons and 20 hexagons. Check Euler’s formula: corners − edges + faces.
10Questions
Cut six strips of thin card about 1.5 cm wide and 41 cm long. Mark ten evenly spaced crossing points on each and label them with the strips they meet. Glue the first into a loop, then weave the others through it one by one, matching the labels and going over and under, gluing each into a loop. Six strips make a round ball about 12 cm across.
Six is the number of great circles that fit together evenly on a ball, each crossing every other one at exactly two opposite points. Their 30 crossings are spread perfectly evenly, and the holes between them are 12 pentagons and 20 triangles, the pattern of a shape called the icosidodecahedron.
It follows the icosidodecahedron, a solid with 12 pentagon faces, 20 triangle faces, 30 corners and 60 edges. All of its edges lie on six great circles, which is exactly what the six strips are. Traditional rattan balls for the game sepak takraw use the same six-strip weave with wider strips.
The biggest circle you can draw on a sphere, like the equator or a line of longitude. Its centre is the centre of the sphere. Any two great circles always cross at two points exactly opposite each other, which is why each strip meets every other strip twice.
For any pattern drawn on a ball, the number of corners minus the number of edges plus the number of faces is always 2. On the woven ball, 30 crossings minus 60 strip segments plus 32 holes gives 2. Leonhard Euler wrote it down in the 1750s.
Safety · Completely safe. Watch out for paper cuts on fresh card edges.
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