The geometry of the familiar soccer ball

There are different designs of soccer ball. The most familiar is the pentagon and hexagon design introduced by the Danish footballer Eigil Nielsen who founded the company SELECT sport. The 2010 official world cup football has a different design.

This is how the pentagon and hexagon football shape would look if it weren’t rounded out.

If you cannot view these pictures, sorry. They are also in the photo album at http://rachelgladstone.angelfire.com/foot32

If you have a craft knife, some stiff card and lots of time, you are very welcome to cut out shapes like those below. However, these are my designs and you may not mass produce or sell them.

Here is a fun sheet about the football and other shapes Archimedean football. As it is an A4 photo of a hand drawn instruction sheet, stored as a pdf file (thank you cute pdf ), it may take a little while to print.

I noticed later that the Archimedean football sheet above includes the Platonic solids in the set of Archimedean solids , which is inaccurate (although perhaps logical). Apologies. I have more up to date versions of this and other polygon related sheets, which I intend to store on www.polywiggle.co.uk soon.

The last picture shows how the tabs are folded and the shapes fitted together. To finish the ball, cut a pentagonal (or respectively hexagonal) shaped hole out of one or more pieces, leaving frames through which you can insert fingers to manipulate tabs. Or, for the last piece, slot the tabs together on the outside of the ball.

Here is a printable pdf file of this hexagon. To print five onto one A4 sheet try five hexagons.

Here is a printable pdf file of this pentagon . To print six onto one A4 sheet try six pentagons.

Warning this is not yet a perfected web page, as sadly I am not a web page designer. This is merely a first draft for those interested. Email rachelgladstone@blueyonder.co.uk to complain.

The small pics of the cards do not print out to scale, but the pdf files should. If you can't print the design to scale but still wish to make a roughly football sized model, try the following: use a ruler and protractor to make one copy of each of the basic pentagons and hexagons. Each side should measure 4 cm (to make a precisely football size model the measurement is a little different). The inner angles between one side and the next are 108 degrees for pentagons and 120 degrees for hexagons.

Now use the measurements on the diagrams above to sketch in the tabs along a 4cm strip, then cut out this strip and draw round it to make the tabs along each edge of your pentagon and hexagon. Cut out and use the pentagon and hexagon with tabs as templates to make the 12 pentagons and 20 hexagons needed for the football (the cutting will take you hours and hours! I hope to manufacture the cut cards cheaply, to save you the trouble). You could also cut out different shapes such as squares, triangles, octagons, to make a range of models.

There are much easier ways to make a football shape. The shape, by the way, is the same as the Buckyball shape (imagine a carbon atom at each corner, with each edge being a bond between atoms). The shape is called a truncated icosahedron. Look up truncated icosohedron nets. A net is a single flat shape that you can fold to make the 3D shape. Or, if you have some money, you might buy a plastic polygon construction kit. I love the one made by GeoAustralia called MiniGeofix. British school children really enjoy playing with Polydron kits, and so do I. My limited experience suggests Polydron frames hold together better than Polydron solid shapes. Lastly, the Association of Maths Teachers make a set of MAT cards which glue together rather effectively with Copydex. Much cheaper than the plastic sets and the results look pretty good, but with 6cm edges and crazy geometric patterns, the effect is not at all football like.

The truncated icosahedron is one of the Archimedean solids (see, for instance, the beautiful book “Platonic and Archimedean solids” by Daud Sutton. Amazing geometry in clear English and accurate pictures. You don’t need to know mathematics or use equations).

The hexagon and pentagon are regular polygons. These are flat shapes that look the same from every corner. Archimedean solids are 3D shapes made from regular polygons that also look the same from every corner. The simplest is made of four equilateral triangles, three meeting at each corner. It is called the tetrahedron. (This is also the smallest Platonic solid). The football shape is one of the larger Archimedean solids and probably the best to use to make a smooth ball, but there are others that could also make balls. Do explore! See also Archimedean football.

Variations of my card designs above can be used to make lanterns, boxes and baskets (I have a special basket edging piece), many different geometric shapes, a few crystal shapes, also models such as chairs, tables, maybe even butterflies and ships.

Rachel Gladstone

June 2010