## Introduction: Revisiting the A4 Parallelepiped

The making of The A4 (DIN A) Parallelepiped was described in my previous Instructable: https://www.instructables.com/id/The-A4-Parallelep.... In it I demonstrated how a particularly proportioned Parallelepiped is embedded in the geometry of a single sheet of A4 paper, whose proportions approximate 1:√2. All papers from this series are sized such that they are related to A0, whose 1:√2 proportional dimensions, rounded to the nearest millimeter, most closely approaches a surface area equal to one square meter, in this case, 841 x 1189. The 1:√2 rectangle is unique in that it is the only rectangle whose proportions remain the same when halved along it's long length. Halving the A0 sheet defines the next one in the series, and halving each successive sheet defines the series. Since the area of A0 approximates one square meter, it follows that A1 sheets are approximately one half of one square meter, A2 are one quarter of a square meter, etc. For more about A series sheets and their relative sizes, see: https://en.wikipedia.org/wiki/ISO_216.

In this lesson, we will review the instructions for the formation of the Parallelepiped, and as we did in that previous Instructable, hinge together multiple units to form space-filling configurations and transformations by creating a "kinematic chain", see https://en.wikipedia.org/wiki/Kinematic_chain . These assemblies will be formed from combinations of groups of three Parallelepiped 'links', hinged together with tape.

Let's proceed.

## Step 1: Assemble Tools and Materials

The materials required are the same as with my earlier work - - at a minimum, A4 paper and tape. A straight edge and scoring tool are recommended for making the diagonal folds. Heavyweight paper such as the 120 gsm used here, helps to make a sturdier model. And of course, any of the A series papers will work, as would any custom sized set of sheets with 1:√2 proportions. And as with my other efforts, there will be no need for measuring tools, nor cutting instruments, just simple folds to full sheets of paper!

As stated in my introduction, we want to link together multiples of three Parallelepipeds, so six sheets of paper are required for this particular exercise.

## Step 2: Fold the Sheets

The fold pattern requires a total of 12 creases. I like to start with three folds which divide the sheet in 4 equal parts along the long length of the sheet. Then I make the three folds that do the same along the short length. Next, I make the two diagonal folds, corner to corner. And last, I do the four diagonals, from center of side to center of adjacent side. See photos.

Forming the Parallelepiped takes some practice, and in past Instructables I have suggested reviewing my earlier Instructable on the making of The A4 Dipyramid as a stepping stone on your path to finding the Parallelepiped. See: https://www.instructables.com/id/UnFolding-the-Mys...

Note to Teachers: As six folded sheets are required for this exercise, all of this folding requires a significant amount of labor, making this, and many of my other Instructables, projects that lend themselves well to group activity, where no one person has to do all of the folding, which I'll admit, does get a bit tedious.

## Step 3: Form and Tape

The photos illustrated here present a visual guide for finding the form, and fixing the figure using adhesive tape. I have applied color coded dots to indicate the areas on the sheet which will become the six finished faces of the Parallellepiped. The top and bottom will consist of identical rhombi (identified with the yellow dots), and the four sides - - a pair of equal parallelograms (blue dots), and a pair that are mirror images of the first pair (green dots). All six of these figures are of equal area, whose surface areas combine to make up three quarters of the total area of each sheet. We know that the six sides are equal just by observing that each is made up of four right triangles, that themselves are all equal. These triangles are the smallest figures created by the fold pattern, each with the approximate proportions: 1:√2:√3. As we have seen, it is easy to calculate that the A4 surface area is approximately one sixteenth of a square meter, therefore the total surface area of our Parallelepiped is three quarters of that, 24 of the total 32 small right triangles.

As we will be assembling linkages of Parallelepipeds in units of three, and then combining those assemblies, repeat to create a total of six Parallelepipeds.

## Step 4: Create Hinge

To create a hinge, or revolute joint, https://en.wikipedia.org/wiki/Revolute_joint, butt two of the 'links' together as shown. Apply tape to create hinging action. Reinforce hinge by applying tape to both sides. Align third link as shown and tape both sides. Repeat to create two hinged groups of three.

## Step 5: Finish Assembly

Hinge the two sets of linkages together as shown. Create closed loop by hinging end to end.

Congratulations, you are done!

Note: Video shows alternate assembly of 12 units. Possibilities abound!

Our ready to fold kits are now available at:

www.etsy.com/shop/Studio20bis

Participated in the

Made with Math Contest