The DIN A5 Rhombic Pyramid

Capitalizing on the inherent proportions of DIN A papers, we will show how to fold and form a Rhombic Pyramid from a single sheet, without the use of measuring instruments or cutting tools.

We will then proceed to show how this pyramid may be used as a building block for the creation of other geometric figures and assemblages.

DIN A paper and adhesive tape are all you'll need for this exercise. Heavyweight 160 GSM A5 paper is used here to achieve a nice sturdy model.

While this fold pattern differs from the one used in my previous Instructables, it generates triangles of the same dimensions and proportions. The inherent 1:√2 proportions of the DIN A paper being used in each case, are what make this so.*

To create our new pattern, we begin by aligning diagonally opposite corners to make the first crease, and proceed by aligning the remaining two corners for crease number 2.

Notice that with these two creases we have found not just the center of the sheet, but the 1/4 and 3/4 points along it's long sides.

Next, fold the sheet in half, along the long length. Making this a valley fold will help when forming the pyramid in Step 2. All of the other folds should be "mountains" relative to this single valley.

Now fold the sheet in half, along the short length.

And finally, fold the four corners from the center of the short sides, to the nearest quarter points.

See photos.

*See Step 6 of my previous Instructable: https://www.instructables.com/DIN-A4-Rhombic-Dode... for my alternate fold pattern for the A5 Rhombic Pyramid.

By pushing the four corners of the sheet simultaneously in and under towards the center, you should find that the pyramid begins to form itself.

Flip the sheet over and you will see four 1:√2:√3 right triangles coming together to form a rhombus. By carefully aligning the edges where these triangles meet, and applying tape, we can fix the rhombic base of our pyramid. By doing so, the pyramid itself becomes rigid.

Dimensioned figure illustrated is from https://mathworld.wolfram.com/RhombicDodecahedron....

As we've shown in the previous Instructable https://www.instructables.com/DIN-A4-Rhombic-Dodec..., as well as others from the series, there are many ways of assembling these units to create interesting geometric figures and "kinematic chains." For another example, see https://www.instructables.com/Kinematics-UnChained... .

Here, we will model the Rhombic Dodecahedron, as well as it's First Stellation, or perhaps more accurately, Escher's Solid. Using a "Net" or pattern, which describes the Rhombic Dodecahedron as if it has been unfolded in such a way that all of it's faces lie in the same two dimensional plane, we can link together 12 of our units, with our rhombic bases substituted for the 2D faces. We do this by hinging their edges together with tape.* Once assembled, it can be seen how hinging them inward towards a center will form the Rhombic Dodecahedron. Hinged outwards, they will form Escher's Solid.

Net illustration is from https://mathworld.wolfram.com/RhombicDodecahedron....

To start, we need 11 more pyramids. Precise folds always lead to a more satisfying result. Take your time.

Owing to the space-filling nature of the pyramids, once they are linked together, the Rhombic Dodecahedron essentially finds it's own form. Repositionable labels (grey dots) are used here for ability to temporarily fix form, and then remove so that we may go on to the next model. Escher's Solid requires a bit more attention, as it surrounds a void equal in shape and volume to that of the Rhombic Dodecahedron. Carefully aligning and linking all of the shared edges, again with the repositionable dots, is used to stabilize the figure and fix it's form.

See photos.

*For hinging I use clear round "Mailing Seals" for a more durable hinge.

Here is a nice little discovery I happened upon while tinkering with this fold pattern.

Rather than taping the 4 right triangles together to form the rhombus, tape as shown, to a create a pair of tetrahedrons that are connected with an integrated, double action "Saloon Door" hinge. The faces of these tetrahedrons will not quite complete themselves, but they are fully defined by their four vertices. See photos.

Enjoy - - and fold your sheets!