Net for Truncated Icosahedron.

Net for Truncated Icosahedron 

32 faces (12 pentagons, 20 hexagons), 60 vertices, 90 edges, Icosahedral (2, 3, 5-fold) symmetry.

Method:-

This net is made in SIX parts:

1. PART 1. (Step - 1) Having chosen your edge length AB on a horizontal line, construct the central pentagon as follows: draw two circles centred on A then B radius AB Draw the vertical axis through the circle intersections C and D to find O (midpoint of AB). Draw a tangent across the top of the two circles, and from the point where this tangent crosses the vertical draw arcs either side radius OA. This gives upper corners E and F of square AEFB. With centre O and radius OE draw a semicircle to cross the horizontal at G and H. AB has now been extended on either side by a Golden Ratio amount. Now draw arcs from A and B with radius AH (=BG). Where they cross on the vertical axis is the upper point J of our pentagon. The other two points are given where the AH and BG ares cross the first circles at L and K. Our pentagon is AKJLB.

Step 1.

2. (Step - 2) On each side of the pentagon construct a hexagon (see Figure) The arrows in the diagram indicate where further sections of the net will be attached.

Hexagon on a line.

Step - 2

3. PARTS 2-5 (Step - 3) Starting with a hexagon (constructed as before) with the same edge length as used in the first drawing, add to the sides one hexagon and two pentagons as shown (Fig.). The pentagons can be more easily drawn this time (see Figure down right) by using the previously drawn pentagon measure with your compasses the length of the diagonal (AJ in Fig.) and draw an arc from each end of the base line. This gives you the apex of the pentagon. Now adjust your compasses to the edge length, and draw an arc from the end of the base line to cross arcs on either side from the apex to find the other two vertices. Finally, one further hexagon is added to the lefthand pentagon to complete this section of the net.

You will need FOUR of these.

Step - 3

Construction of pentagon using arcs of known measurements.

4. PART - 6. (Step - 4) is similar to the previous ones, but with an extra pentagon on one end (Fig.)

Step.- 4.

5. Add tabs, and cut and score lines, as shown. 

6. Cut out along the solid lines: score along the dashed lines. Fold upwards along the scored lines Glue carefully and progressively (tabs inside) - one of two tabs at a time. Clean up and decorate as you wish.👍

Illustration of the six parts of the net drawn out on card and cut out.

Note:

Until 2006 the official World Cup football was made in the shape of a Truncated Icosahedron.

Illustrations of four of the six parts of the net glued (top left); the figure partially assembled (top right); further assembled (bottom left); and complete (bottom right).