## Icosahedron-Tri

Materials needed

Construction paper - preferably sulfide, 12”x18”
Pattern: Make mountain folds on the blue lines. Ignore the light gray lines; they only indicate the structure of the strip. Cut on the black lines. This pattern can be downloaded.

Scissors
Cardboard backing
Push pins
Sharp table edge
Paper clips
Stapler

The solid

Schläfli symbol: 3.3.3.3.3
Conway Symbol: Solid: I; Plait: jI
Type of plait: triangular
Plaiting pattern: Pentyl (6 strips each 10 quadrilaterals long)
Size Ratio: 3.294556

The icosahedron is the solid consisting of twenty triangular faces. Its name is Greek for "twenty-sided". It is used as a 20-sided die and appears in the logo of the Mathematical Association of America.

Pattern

There is one pattern piece for this model. The model requires six of these pieces, which should be 11 quadrilaterals long, 10 plus one for the tuck-in piece. A quadrilateral consists of two obtuse triangles connected at an edge where a fold occurs. A triangle is sufficient for the tuck-in, but a full quadrilateral will make a more sturdy solid. The best color scheme is six different colors. This is because five pieces meet at each vertex and need to be of different colors, and the sixth piece girds the equator and touches the other five.

Divide the size that you want by the size ratio above to get the size to enter in the Weaving Paper Polyhedra software program to produce the patterns.

This solid is one of the easier big solids to make, since its pieces are identical to those of Octahedron-Tri and Tetrahedron-Tri, but longer. If you are a beginner and want an impressive result, choose this model.

This solid is especially attractive but looks more like a solid with five-pointed stars at each vertex, rather like an embossed edition of Pentyl, than like an icosahedron. This is because the quadrilaterals fold in the middle to form two triangles, and the folds are hard to bring out. If you want to bring out the icosahedronness of this solid, make sure the folds are sharp. This solid is especially suited to the Christmas tree, either as an ornament or as the top.

1. Cut out the pieces

Lay the backing on a table. The purpose of this backing is to absorb the pricks from the push pin. Lay the required number of sheets of construction paper on the backing, one on top of the other. Lay the pattern on top of the construction paper stack.

Prick each corner of the pattern with the pin hard enough to go through all the pieces of construction paper.

Using the pricks as a guide, cut the pieces out from the construction paper. First cut around the pricks so that you don’t have to deal with the entire sheet of construction paper. Cut from prick to prick. Using a straightedge to draw lines between pricks to determine where the cut should go will result in a more accurate cut but will also result in a pencil mark to erase.

Several sheets can be cut at the same time. Staple the sheets together, being generous with the staples. Be careful that the sheets don’t wander when you cut them - that could result in inaccurately cut pieces.

2. Fold the pieces

Lay the pattern on a table. Lay a piece next to it so that it is oriented the same as the table pattern. Fold on the inside according to the solid lines in the pattern. Use a sharp table edge or ruler edge to make a clean fold. The folds should cause the strip to curve upward.

Fold each strip the same way. Folding two or three at once will save time but may result in inaccurate folds. Folding the strips so that they are oriented the same as the table pattern, so that you can stack the strips on top of each other, will ensure that the strips will fold together so that the tabs will tuck under, not over.

3. Assembling the model

This model is formed by weaving the strips together. A strip goes over another strip and then under the next one, continuing the under and over pattern around the model.

Two strips overlap at a quadrilateral, a polygon with four sides. Such a polygon has two pairs of opposing edges. One strip goes over one pair of opposing edges, and the other strip over the other pair.

Therefore, the quadrilaterals on each strip alternate between under and over. Laid out flat, the strip looks like this:

under, over, under, over, under, over, under

The tabs tuck under a quadrilateral from another strip, a little like closing the cover on a cardboard box. So when assembling the solid, make sure that the tabs go under. When introducing a new strip, count from the crossing point to the end. If the strip is inserted under another one, for example, count "under, over, under…" until you reach the end. If you get "over" when you reach the end, the strip is inserted incorrectly.

The weaving operation leaves a result consisting only of the strips of construction paper, but paper clips are needed to form the model. The paper clips are removed when the model is completed.

a. Fasten two strips together. Align two strips so that they overlap at a quadrilateral, which in this solid is skew and consists of two obtuse triangles connected at a fold that marks the edge of the solid. Make sure that the end flaps go under, rather than over. To do this, count from the overlap to the end of the piece. If the piece goes over the overlap, count over, under, over, … until you get to the end. If you get over at the end, you matched them up wrong. Both strips must be set up to go under at the end.

One way with this model is to overlap the sixth quadrilateral of one piece over the fifth quadrilateral of another one.

b. Insert three more pieces to form a five-pointed star at a vertex. Up to now the assembly is the same as for Octahedron-Tri. Where it now differs is that you weave in pieces so that five pieces come together at a vertex instead of four.

At times pieces may get buried under and then you have to bring them out. At times you may find a piece ends where it should continue and weave with other strips, so that you may have to skip over a weave and get it back later.

Insert the remaining piece and the solid should weave and weave until it closes up to form the icosahedron.