## Introduction: Kinematics (Un)Chained

In this Instructable we return to our basic unit, the DIN A4 Dipyramid. This time we will assemble groups of four, then hinge the groupings to create yet another illustration of the space-filling nature of our building block.

At the same time, we will be creating for ourselves another fun little toy to play with!

## Supplies

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 the final model requires 4 groupings of the 4 Dipyramids, 16 sheets of paper are needed.

For more on this proportional relationship and it's ties to the Dipyramid, see my previous Instructable:

## Step 1: Fold the Sheets

I have described the method for folding the sheets in great detail in (Un)Folding the Mysteries of A4 Paper, Steps 2-6 https://www.instructables.com/id/UnFolding-the-My... The basics are as follows:

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 3 folds that do the same along the short length. Next, I make the 2 diagonal folds, corner to corner. And last, I do the 4 diagonals, from center of side to center of adjacent side. Prepare all 16 sheets in this manner. See photos.

## Step 2: Form the Dipyramids

Forming the figure is illustrated in the photos shown here. Again, for more detailed instruction, see (Un)Folding the Mysteries of A4 Paper Steps 8-10.

## Step 3: Assemble Cluster

This too, closely follows the instructions for assembling the Rhombic Dodecahedron described in (Un)Folding the Mysteries, beginning at Step 11. For this exercise we can follow those detailed instructions through Step 13.

Briefly, the idea is to create a hinge, or revolute joint, https://en.wikipedia.org/wiki/Revolute_joint by butting two of the Dipyramids (links) together as shown. Then, apply tape to create hinging action. Reinforce hinge by applying tape to both sides. Repeat to create 4 hinged groups of 4. Then, fix each cluster by taping, end to end, as shown.

See photos.

## Step 4: Hinge the Clusters

Butt and tape hinged clusters as shown. Close the loop to create a kinematic chain https://en.wikipedia.org/wiki/Kinematic_chain.

Well Done!

Our ready to fold kits are available at:

www.etsy.com/shop/Studio20bis

For those of you who may want to delve deeper into the mathematics, I recommend the following papers by Guy Inchbald, each with some very nice CAD drawings relating to our model:

Five Space-Filling Polyhedra, The Mathematical Gazette 80, No. 489, November 1996, p.p. 466–475.

The Archimedean Honeycomb Duals, The Mathematical Gazette 81, No. 491, July 1997, p.p. 213–219.

They can be found at his site:

http://www.steelpillow.com/guy/published/index.htm...

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