Not to be upstaged by what some might consider the ‘highbrow’ nature of A4 paper, it turns out that the lowly 9” x 12” sheet of Construction Paper, found in kindergarten classrooms since the late nineteenth century*, has some fine inherent geometric qualities of its own, relating it to our favorite polyhedra: the regular dipyramid with height (h) equal to the length of one of its sides (a): h=a. This time it is the dipyramid with a hexagonal base which can be found embedded in a single sheet.

In my first Instructable: https://www.instructables.com/id/UnFolding-the-Mys... we saw how such square based dipyramids are embedded in the 1:√2 proportions shared by all A series papers. And then, in: https://www.instructables.com/id/Dipyramids-UnMaske... , we learned how to find the proportions of the rectangle required for folding and forming any regular dipyramid whose height (h) is equal to the length of one of its sides (a): h=a. To do so, we found that the number of sides (n), the length of side (a), and the circumradius of the base (R), were all that were required for determining the dimensions of that rectangle, and thus its proportions. We proceeded by doing the calculations for a five sided base, and a step by step demonstration of how to make the model.

While the demonstration stopped there, illustrated were models of hexagonal and octagonal versions of the dipyramid. In doing the calculations for the hexagonal, I found that the required rectangle had the dimensions: 210mm x 280mm. Recognizing this as a 3:4 ratio led me to wonder about the availability of standard size, off-the-shelf sheets with these proportions. Though not as prevalent as 8 1/2” x 11”, I found that 9” x 12” sheets are not uncommon in the States, and construction paper, in particular, is available wherever you go.

So in this Instructable, we will review our technique for creating the dipyramid, this time in its hexagonal form. And for fun, we will proceed by demonstrating how to assemble a modified Truncated Icosahedron, using 20 of the models. See: http://mathworld.wolfram.com/TruncatedIcosahedron...

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## Step 1: Tools and Materials

As the 3:4 proportions are a given, we know that no trimming will be required, but we will have to divide the short length of our sheet in six equal parts, per our previous analysis in 'Dipyramids (Un)Masked'. Therefore we will use a ruler, plus our standard set: tape, straightedge, scoring tool; and 20 sheets of construction paper.

## Step 2: Score and Crease

Each sheet is to be scored and creased, as shown. Orthogonal creases to create a 6 x 4 array, then diagonals, to form the diamond pattern. See my previous Instructables cited in the introduction, for detailed reasoning and instructions.

Note: Diagonals do not run corner to corner.

## Step 3: Fold Form and Tape

Fold, form, and tape each sheet as shown.

See my previous Instructables for detailed instructions.

## Step 4: Begin Hinged Connections

Apply tape as shown. Remember to tape both sides at each hinge for stability and durability.

Repeat to create 10 pairs.

See my previous Instructables for detailed instructions.

## Step 5: Continue Hinged Connections

Tape as shown.

See diagram, top of web page: http://mathworld.wolfram.com/TruncatedIcosahedron....

Note: The pentagons shown in diagram will become voids in our model.

## Step 6: Continue Assembly

Continue taping, as shown.

## Step 7: Finish Assembly

Tape until all pentagonal voids are defined.

Congratulations!

Participated in the

Paper Contest 2018

## Discussions