Revisiting the A4 Dipyramid and Finding the First Stellation of the Rhombic Dodecahedron
Introduction: Revisiting the A4 Dipyramid and Finding the First Stellation of the Rhombic Dodecahedron
Towards the end of my previous Instructable, https://www.instructables.com/DIN-A4-Rhombic-Dodec..., we saw how twelve A5 Rhombic Pyramids could be hinged together following the "net" (template) for a Rhombic Dodecahedron, and then folded inward to form the figure, where the base (bottom) of each pyramid becomes one of the rhombic faces of the dodecahedron.
We also noted that two of these A5 pyramids, sharing a common rhombic base, are identical in shape to our standard A4 Dipyramid. Given these observations, we will now proceed by linking twelve of these rhombic based A4 Dipyramids using that same net, and see how they can be manipulated to form The First Stellation of the Rhombic Dodecahedron.*
*https://mathworld.wolfram.com/RhombicDodecahedronS...
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. Like my earlier efforts, there is no need for measuring tools or cutting instruments, just simple folds to full sheets of paper!
Twelve sheets of A4 paper are required for this particular exercise.
Step 1: Fold the Sheets
Those of you who have been following along will know the fold pattern very well, but as is my habit, I will review.
The fold pattern requires a total of 12 creases. 518
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.
Step 2: Form and Tape
As our Rhombic Dipyramid and Square Dipyramid are one and the same, we can use our Square Dipyramid technique, described in my very first Instructable, https://www.instructables.com/UnFolding-the-Myster....
The photos show how to find the form, and where to apply the tape. Repeat to make a total of twelve Rhombic Dipyramids.
Step 3: Create Hinge
Use the Rhombic Dodecahedron "net" for the hinging pattern.
Remember to align carefully and apply tape to both sides at each hinge.
See photos.
The net can be found here:
*https://mathworld.wolfram.com/RhombicDodecahedron....
Step 4: The First Stellation of the Rhombic Dodecahedron
Hinge inward as shown to create the First Stellation of the Rhombic Dodecahedron, also known as "Escher's Solid".*
Continue to see Bonus.
Step 5: An Unhinged Bonus
The video shows an alternate assembly of the same twelve units.
Unhinged, these last photos then illustrate how the volume of The First Stellation of the Rhombic Dodecahedron (twelve dipyramids) is equal to twice the volume of it's associated Rhombic Dodecahedron (six Dipyramids each).
Be safe out there - - and if you find yourself cooped up with nothing to do - - fold your sheets!
Our ready to fold kits are available at: www.etsy.com/shop/Studio20bis
* https://mathworld.wolfram.com/EschersSolid.html. 518