## Introduction: (Un)Folding the Mysteries of A4 Paper -- There's Some Pretty Far Out Geometry in There

By design, 'A' Series paper*, not widely available in the United States but ubiquitous throughout most of the world, lends itself to all manner of geometric 'discoveries' just by folding. This instructable is meant to be an introduction to the vast possibilities embedded in a single sheet. The set of instructions leads to a very specific model, taking you step by step to the formation of a Square Dipyramid**. The instructions and photos are meant to be enough to get you there without frustration, but with opportunities for making your own discoveries along the way. It proceeds to show how multiples of the model can be assembled, in building block fashion, leading to our final Rhombic Dodecahedron***. All of the work is made from full sheets of A4 paper, and requires no measuring or cutting, nor is it necessary to perform any mathematical calculations.

I hope you enjoy!

* https://en.wikipedia.org/wiki/Paper_size

** http://mathworld.wolfram.com/Dipyramid.html

*** http://mathworld.wolfram.com/RhombicDodecahedron....

See also http://mathworld.wolfram.com/SquarePyramid.html for illustration of special case Square Pyramid where h = a/2. Anything look familiar?

## Step 1: Assemble Tools and Materials

Required:

Six sheets of A4 paper -- I'm using 100 gsm kraft paper, but in fact any 'A' series paper will work.

Tape

Optional:

Double-sided tape

For diagonal folds, I recommend scoring the fold using a straight edge and bookbinder's scoring tool or the equivalent. When using a scoring tool, the heavy weight paper is advised.

In addition, the flat side of the scoring tool, or the barrel of a pen, comes in handy when forming creases, though a finger will do.

## Step 2: Vertical Folds - Part 1

Fold first sheet in half along long length, making sure to precisely align short sides, one on top of the other.

Crease.

## Step 3: Vertical Folds - Part 2

Fold back short side of top half of sheet to precisely align edge with center fold.

Crease.

Flip over and repeat to create accordion fold with four equal panels.

Unfold.

## Step 4: Horizontal Folds

Repeat steps 2 and 3 along short length of sheet.

Unfold to reveal sheet divided into 16 equal rectangular divisions.

## Step 5: Diagonal Folds to Corners

Score then Fold diagonally from corner to corner.

Crease, unfold, and repeat, at opposite corners.

Unfold.

## Step 6: Diagonal Folds to Center of Sides

Score then fold diagonally from center of a long side to center of an adjacent short side.

Crease and unfold.

Repeat three more times to all center of side to center of side adjacencies.

Unfold.

## Step 7: Repeat

Repeat Steps 2-6 on all (total of six) sheets.

## Step 8: Manipulate Folds

Take Sheet 1 and manipulate folds.

Through trial and error, you should be able to find your way to our desired shape, known as a 'Square Dipyramid'.

Use photos to help guide you.

## Step 9: Assemble Square Dipyramid

Apply tape as shown in image 1. Rotate 180 degrees and tape opposite seam, image 2. Bring the four remaining surfaces in, to form the bottom pyramid, and tape as shown, image 3.

Tip: Make sure to carefully align corners and butt edges before applying tape.

## Step 10: Repeat

Repeat steps 8 and 9 -- all sheets -- to create six Dipyramid 'building blocks'.

## Step 11: Assemble - Part 1

Butt two of the Dipyramids together and apply tape as shown.

This will create a hinge between the two.

## Step 12: Assemble - Part 2

Place pair from previous step. Butt and apply tape to two more Dipyramids, one to each side, as shown

Tip:

Check out the hinging action of this assembly before proceeding -- it makes a nice little toy.

## Step 13: Assemble - Part 3

Form the 'square inscribed by a square' configuration as shown, and apply tape to fix position.

Tip:

You may want to use double sided tape for this and the final steps in order to keep tape from appearing on the exterior surfaces of the final model.

## Step 14: Finish Model

Position and tape top and bottom Dipyramids, as shown.

Congratulations! You have made your very own Rhombic Dodecahedron.

Rhombic Dodecahedron: took a few day. Then it folded the four correct triangles to a pyramid. Then the other four folded making the double. Do five more modules and there was the Rhombic Dodecahedron. Now try to draw in 3-d cad.

That's pretty fun to see all of the things you can make :)