The A4 Parallelepiped - - It'll Blow Your Mind

A Parallelepiped* is a three-dimensional figure formed by six parallelograms, and lo and behold, there is a particularly nice one that can be modelled from a single sheet of A4 (DIN A) paper. As with the Square Dipyramid and other figures we've looked at in my earlier Instructables, this figure can be formed using our standard fold pattern, without any need to measure, cut or perform calculations. In addition, like the Dipyramid, the Rhombic Dodecahedron, and the other modified figures we've looked at, our Parallelepiped is a 'Space Filling Polyhedron'**. So as we did with the Square Dipyramid, after learning how to form one, we will look at an interesting way of assembling multiple units.

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

http://mathworld.wolfram.com/Parallelepiped.html

** http://mathworld.wolfram.com/Space-FillingPolyhedr...

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.

Step 2: Fold, Form and Tape

After completing the fold pattern, I like to form the Dipyramid, and then modify it to find the Parallelepiped form. Our model will consist of two rhombi, top and bottom, and four sides that are similar parallelograms. The Rhombi are formed by taping their two halves together along their long axes. Continue to tape as shown. See photos.

For detailed instructions and tips, see my earlier Instructable, (Un)Folding the Mysteries of A4 Paper:

https://www.instructables.com/id/UnFolding-the-Mys...

Step 3: Apply Tape to Create Hinge

Assemble three Parallelepipeds. Align as shown and use tape to create hinged connections. The photos illustrate how after hinging, the units will begin to spiral when hinged in this manner.

See my earlier work, Mysteries of the (Di)Pyramids https://www.instructables.com/id/Mysteries-of-the-... for detailed instructions on hinging.

Step 4: Repeat Steps 2 and 3

Repeat steps 2 and 3 to create two more hinged groups of 3 Parallelepipeds. Align as shown.

Step 5: Continue Assembly

Tape together the three assemblies as shown to form array of nine units. Roll up the array to form the spiraling Parallelepipeds . It is like a rope with three strands. The assembly takes the form of a Right Regular Hexagonal Prism, albeit with irregular bases, top and bottom. The Hexagon is identical in its dimensions to the one seen in an orthogonal view of our Rhombic Dodecahedron from previous Instructables (see photo). This Right Regular Hexagonal Prism, like a Honeycomb Cell, is also a Space-Filling Polyhedra.

See https://en.wikipedia.org/wiki/Triangular_prismatic...

https://en.wikipedia.org/wiki/Rhombic_dodecahedral...

Step 6: Conclusion

Speaking of Honeycomb Cells, we will finish our model by topping it off as the bees might do it. To do this we must revisit my earlier Instructable, Rhombic Dodecahedron 3-Ways https://www.instructables.com/id/Rhombic-Dodecahed... to see how we modelled our Heptahedron (seven-sided polyhedra). Once you have reviewed the instructions, make three Heptahedra and tape together, as shown. Place assembly on top of Prism. See photo.

This final step was inspired by the work on Rhombic Dodecahedrons and Honeycomb cells done by Roberto Cardil, at his wonderful Website MatematicasVisuales. See http://www.matematicasvisuales.com/english/html/ge...

Thank you, Roberto, for all of your fine work.

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