DIN A4 Rhombic Dodecahedron - Again!

In this exercise we will form a new figure from the standard DIN A4 fold pattern that I introduced in my first Instructable: https://www.instructables.com/UnFolding-the-Myster...

Then, we will assemble multiples of this new figure to form a Rhombic Dodecahedron. This assembly requires 4 of these new forms.

While the dodecahedron's edges will be fully defined, four rhombic pyramidal voids will remain.

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.* Like my earlier efforts, there is no need for measuring tools or cutting instruments, just simple folds to full sheets of paper!

As stated above, 4 sheets of A4 paper are required for this particular exercise.

*see my Instructable: https://www.instructables.com/Dipyramids-UnMasked/ for a detailed analysis of this special rectangle and the fold pattern associated with it.

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.

Finding the form of our new shape will take some effort, so as in past Instructables I suggest reviewing my earlier Instructable on the making of The A4 Dipyramid as a stepping stone on your path to finding this new form. This will also provide more detailed folding instructions, emphasizing the need for precision folds to achieve tight fitting (space-filling) assemblies.

See: https://www.instructables.com/id/UnFolding-the-My...

The photos illustrated here present a visual guide for finding the form, and fixing the figure using adhesive tape.

Before applying the last piece of tape, notice that we've modelled two identical rhombic based pyramids hinged along a common edge. We will revisit this discovery in Step 5 of this Instructable.

After the final piece of adhesive tape is applied, repeat to create a total of 4 of these figures.

Create two hinged pairs, as shown. Remember to align carefully and apply tape to both sides at each hinge.

Hinge the pairs together as shown.

Form the Rhombic Dodecahedron and fix with tape. One piece is all that is required. See photos.

In my previous Instructable: https://www.instructables.com/Kinematics-UnChained... I acknowledged the work of Guy Inchbald. Here, I reproduce a chart from his article "The Archimedean Honeycomb Duals", which can be found on his website: http://www.steelpillow.com/guy/published/index.htm.... In this chart we recognize some of the geometries imbedded in DIN A paper that drive my Instructables. I present the chart not just for its graphic clarity, but also so that we may borrow from its nomenclature.

The figures we want to highlight are:

The Rhombic Docdecahedron

The Rhombic Hexahedron (a Parallelepiped)

The Rhombic Pyramid

And, of course, what he calls The Oblate Octahedron, which is the same as our basic figure, The DIN A Dipyramid.

The rhombi associated with the first three named above, are the ones found in our fold pattern. It's proportions are defined in the diagram I've borrowed from Mathworld.* The surfaces of the first two of these, dodecahedron and hexahedron, are made up exclusively of these rhombic faces. The third, the pyramid, has the rhombus as its base, and 4 similar isosceles triangles, each one equal to one half of the rhombus, completing the figure. This is the figure that we found two of, hinged together, that we called attention to in Step 2 of this Instructable.

We can see from Inchbald's flow-chart how a pair of these pyramids, sharing their rhombic bases, will form a dipyramid, what he calls The Oblate Octahedron. This figure is in fact identical to our own square based Dipyramid, and is shown as such in the chart. Turned 90 degrees, with the rhombus in the horizontal plane, it reads as The Rhombic Dipyramid. See photos.

The form we modeled today from our single sheet of A4 paper consists of a cluster of two of these Rhombic Pyramids, sharing sides, not bases. This creates a figure forming two thirds of the Rhombic Hexahedron seen in the chart. As the chart tell us, the complete hexahedron requires 3 such pyramids, thus the voids in our model.

I will now show how we can model a single Rhombic Pyramid from one sheet of A5 paper to complete the hexahedron. And finally we will see how four of these would fill the voids found in our Rhombic Dodecahedron.

*the diagram is from https://mathworld.wolfram.com/RhombicDodecahedron....

To maintain the proper scale for our model, the A5 fold pattern is as follows:

Make three folds to divide the sheet in four equal parts along its long length.

Fold in half along the short length.

Fold diagonals from center of long lengths to corners.

Form and tape as shown.

Repeat to make a total of 4 Pyramids.

Fill the voids.

The photos now show 4 complete Rhombic Hexahedrons, linked, then folded, to form the dodecahedron.

One can easily see how 12 of these individual units, The A5 Rhombic Pyramid, could be linked to form our Rhombic Dodecahedron. The pattern for it's net, shown here, would serve equally well as a hinging diagram, creating the same "space-filled" model. See: https://mathworld.wolfram.com/RhombicDodecahedron.... and https://mathworld.wolfram.com/Net.html.

Note: If you do not have access to A5 sheets, you can always cut your A4 paper in half. I won't tell anyone, but as always, be precise!

With two more Hexahedrons, you can reassemble to create the linked loop shown. (Bonus #2)

Stay home and keep folding!

Our ready to fold kits are still available at:
www.etsy.com/shop/Studio20bis