Today we will look at three different ways of modelling Rhombic Dodecahedrons of equal dimensions, using whole folded sheets of A4 paper. There will not be any calculations, measuring, or cutting required for any of our work. My first Instructable, (Un)Folding the Mysteries of A4 Paper, demonstrates the first of the three methods. It shows how, by modelling six dipyramids, each from a single sheet of A4, the desired Rhombic Dodecahedron can be assembled. Today we will see how Rhombic Dodecahedrons of the same dimension can also be assembled from five sheets of folded A4 paper, and again from just four sheets, always using the same fold pattern.

Step 1: Assemble Tools and Materials

The materials required are the same as with the original, at a minimum, A4 paper and tape. A straight edge and scoring tool are recommended for making diagonal folds, and double stick tape can be used for a cleaner look (eliminating seeing exposed tape on the surface of the model).

Step 2: Version I

This set of instructions does not just walk you through how to make the first version, but also sets you up with the basic skills and techniques for creating the next two. There are a number of additional references, and helpful hints included along the way

Throughout, I will be using contrasting colored sheets to distinguish parts from the whole.

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

Step 3: Version II

Version II is formed from five sheets, one of which takes the form of the same dipyramid used in Version I. The other four are each heptahedron (seven-sided polyhedra) consisting of four faces that are isosceles triangles, two that are similar parallelgrams with mirror images of each other (these surfaces get buried in the final model), and one rhombus. The shape is easily formed by manipulating our standard dipyramid.

To form the heptahedra, proceed up to Step 9 of the dipyramid instructions. Apply the first piece of tape to the bottom pyramid as shown in Step 9 and then proceed as follows. Open the other side of the bottom pyramid to form the heptahedron and apply tape as shown. Assemble the five parts to create Rhombic Dodecahedron - Version II. See photos.

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

Step 4: Version III

This version is created from four identical quarter sections of the whole Rhombic Dodecahedron figure, also easily derived from a simple manipulation of our dipyramid model. This shape is a hexahedron, made up of two trapezoids, two rhombi, and two isosceles triangles. In this case, it is the trapezoids that get buried in the model.

To form the hexahedra, proceed up to Step 9 of the dipyramid instructions. In this case both sides of the bottom pyramid are opened to form the hexahedron. Apply tape as shown. Assemble the four parts to create Rhombic Dodecahedron - Version III.

See photos: