Making a Soccer Ball

A soccer ball, made from 12 pentagons and 20 hexagons, is a good example of an object that would be very difficult to construct by entering the coordinates of all its vertices. The geometry involved in calculating all those coordinates is way beyond the patience of most mere mortals. Not quite so difficult is a dodecahedron made out of 12 pentagons.

Fortunately there is an easier way to make objects where the face shapes are known in advance. In L3DD we call this the ‘Tilt-Up Mechanism’. It is essentially the same as the method you would use to construct objects face by face out of cardboard and adhesive tape.

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Starting with a horizontal fixed face (a pentagon, for a soccer ball) you join two horizontal faces (hexagons for a soccer ball) to the pentagon by one edge each. These are known as ‘hinged faces’. The hinged faces are then tilted upwards until they share a common edge. This is the only occasion the tilt-up mechanism is needed. The rest of the ball is built with the help of L3DD’s rotational symmetry capabilities.

Building a soccer ball is an excellent demonstration of the power of L3DD. The steps in producing a soccer ball are:

1.	Click on the New button to clear the decks for a new project.

2.	Select the Y Rotation Axis command and specify a 5-fold rotation axis.

3.	Enter a new vertex at X= 50, Y = 20, Z = 0 with the Add New Vertex command. Five new vertices are instantly generated.

4.	Click on the background and then select each of the vertices in counterclockwise order to create a new pentagonal face parallel to the groundplane.

5.	Press ‘E’ to turn on edge handles. Select an edge.

6.	Select Add Hinged Face and choose six edges in the dialog box. Five hexagonal faces are generated around the pentagonal face.

7.	De-select the Y Rotation Axis.

8.	Select three hexagons with the highest face numbers and delete them. Select Clean Up to remove stray vertices and renumber everything.

9.	Select the vertex common to all three faces then use the Join Hinged Faces command. The two hexagons are now joined together with the pentagon.

10.	Set a 5-fold Y rotation axis again, select face 2 and click on the Apply Symmetry button to make 5 hexagonal faces around the pentagon.

11.	Select face 2. Use the Rotate to Bottom and Centre on Y commands.

12.	Establish a 3-fold Y rotation axis. Select the pentagon and 2 adjacent non-horizontal hexagons. Your soccer ball should now look like the illustration to the right.

13.	Use the Apply Symmetry command to generate nearly half the soccer ball.

14.	Select face 1, Rotate to Bottom and Centre on Y , then establish a 5-fold Y rotation axis.

15.	Repeat procedures like those in 10 to 13 until the soccer ball is complete.

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Use ‘Center Object’ to center the soccer ball on the origin.actions that rotate selections or the whole object within the axes (e.g. Rotate to Bottom) introduce small trig errors each time they are used. Minimize their use to reduce cumulative errors. The faces with the lowest face numbers are the best for rotating to the bottom and for generating other faces via a rotation axis. If you select the entire soccer ball and use the Face(s) Flat command you will find out just how good your soccer ball is. Faces that are flat or reasonably flat remain blue. Those that are still flat within current tolerance turn orange, and those that are outside the tolerance turn red:

Your ball may have a few orange faces, but it is possible to construct a ball that is entirely blue. If your ball has orange faces you may be able to fix it by selecting the whole ball, using the Spherify command, and retesting for flatness. Failing that, the Cancel Face Colors command will restore all faces to blue.

With the advent of the ‘Chamfer Object’ command in Version 3.20 of L3DD, an alternative way of creating a soccer ball has been serendipitously discovered. If you select the Dodecahedron from the Object Menu you can create a soccer ball by chamfering all its vertices in one hit. First select the whole dodecahedron with the ‘Select-All’ button, and then use the ‘Chamfer Object’ button in the Boxatrix Toolbar. By selecting ‘Vertices’ and entering as chamfer depth of 23.5, a near-perfect soccer ball is created. A perfect soccer ball can be achieved by further trial-and-error adjustment of the chamfer depth. Its interfacial angles are identical to those of a soccer ball made with the tilt-up mechanism described above.