Introduction: Eighty Pairings - A4 Parallelepiped (Hommage À Sol LeWitt)
My past Instructables have largely focused on how several polyhedra, but in particular, a Dipyramid
https://www.instructables.com/id/Dipyramids-UnMask...
and a Parallelepiped
https://www.instructables.com/id/Revisiting-the-A4...
can be formed from full sheets of DIN A4 paper. We have seen the space-filling nature of these figures, and have examined interesting ways of combining multiple units, forming for example, The Rhombic Dodecahedron
https://www.instructables.com/id/UnFolding-the-Mys...
The Right Regular Hexagonal Prism
https://www.instructables.com/id/The-A4-Parallelep...
and a variety of Kinematic Chains, using the figures, or clusters of them, as links in the chain.
https://www.instructables.com/id/Mysteries-of-the-...
https://www.instructables.com/id/DipyramidParallel...
https://www.instructables.com/id/Kinematics-UnChai...
Each new effort began by reviewing the materials and supplies required; described the fold pattern, the techniques used to form figures, and showed us how to assemble, hinge, and manipulate the models. Our future studies will continue to focus on creative combinations of the basic units: Dypyramid (DP) and Parallelepiped (PP), but from here out, I will assume that those who have followed along are familiar with these basics, and those who are new to my work, or need to refresh, will refer back to the earlier Instructables referenced above.
In this Instructable, we home in on the PP, and systematically review all of the ways of linking them together in pairs. In doing so, we will allow for a better understanding of the characteristics of complex chains of multiple units.
Supplies
Required: DIN A4 Paper and Tape.
Recommended: Straightedge and Scoring Tool
Step 1: Nomenclature
For the purposes of this Instructable, the Top View of the Parallelepiped will be defined as the view showing no tape, and positioned with the Top Rhombus (TR) facing up and side parallelograms sloping down from the bottom half of Top Rhombus. Given this orientation, we can name the parallelogram sloping to the right of center, the Top Right Parallelogram (TRP), and the one to the left, the Top Left Parallelogram (TLP).
The Bottom View will expose the taped sides, and will be positioned with the Bottom Rhombus (BR) facing up and the side parallelograms sloping down from the bottom half of the Bottom Rhombus. And as above, we will name the bottom parallelograms relative to this view, calling the one right of center, the Bottom Right Parallelogram (BRP), and the one to the left, the Bottom Left Parallelogram (BLP).
In the photos, I have applied colored dots, for a visual aid, to help us
understand this naming of the six different sides of the DIN A4 Parallelepiped:
Top Rhombus (TR): Blue
Top Right Parallelogram (TRP): Red
Top Left Parallelogram (TLP): Yellow
Bottom Rhombus (BR): Black
Bottom Right Parallelogram (BRP): Green
Bottom Left Parallelogram (BLP): Brown
Step 2: Pairing the Parallelepipeds
Our PP, like all parallelepipeds, is a figure with 6 faces and 12 edges. In our case, it has edges with two distinct lengths. The edges of the rhombic faces, TR and BR, are 91 mm long, rounded to the nearest mm (Length A), so 8 of the edges are of this length. Its 4 additional edges, the long lengths of the 4 parallelograms, TRP, TLP, BRP, and BLP, measure 105 mm, (Length B). (See https://www.instructables.com/id/Dipyramids-UnMask... for our calculations used to find these lengths).
We will now proceed to systematically review all of the ways of linking together two A4 Parallelepipeds, when they are joined by linking edges of equal length, either A to A, or B to B. The photos above illustrate one example of each. The total number of combinations are 64 for Length A, and 16 for Length B.
The next steps will systematically illustrate each of these pairings, beginning with Length A pairs.
Step 3: Length-A: Sets 1 to 4
For Length A pairings, 16 sets of 4 pairings shall adhere to the following conventions in their photographic representation:
PP1 will remain fixed in place, and PP2 will move relative to it, to show which edges are being linked.
Each photograph will represent one set of examples showing the four ways in which one edge of PP1 can be paired with the four edges of any one face of PP2, rotating clockwise from one to the next.
Set 1 illustrates the pairings of the top left edge of TR/PP1, to the four edges of TR/PP2 .
Set 2 shows the four edges of TR/PP2, to the top right of TR/PP1.
Set 3 shows bottom left and right TR/PP1, to top left and right TR/PP2.
Set 4 shows bottom left and right TR/PP1, to bottom left and right TR/PP2 (seen in rotated position).
Step 4: Length-A: Sets 5 to 8
Set 5 illustrates the pairings of the top left edge of TR/PP1, to the four edges of BR/PP2 .
Set 6 shows the four edges of BR/PP2, to the top right of TR/PP1.
Set 7 shows bottom left and right TR/PP1, to top left and right BR/PP2.
Set 8 shows bottom left and right TR/PP1, to bottom left and right BR/PP2 (seen in rotated position).
Step 5: Length-A: Sets 9 to 12
Set 9 illustrates the pairings of the top left edge of BR/PP1, to the four edges of BR/PP2.
Set 10 shows the four edges of BR/PP2, to the top right of BR/PP1.
Set 11 shows bottom left and right BR/PP1, to top left and right BR/PP2.
Set 12 shows bottom left and right BR/PP1, to bottom left and right BR/PP2 (seen in rotated position).
Step 6: Length-A: Sets 13 to 16
Set 13 illustrates the pairings of the top left edge of BR/PP1, to the four edges of TR/PP2.
Set 14 shows the four edges of TR/PP2, to the top right of BR/PP1.
Set 15 shows bottom left and right BR/PP1, to top left and right TR/PP2.
Set 16 shows bottom left and right BR/PP1, to bottom left and right TR/PP2 (seen in rotated position).
Step 7: Length-B: Sets 17 to 20
For Length B pairings, 4 sets of 4 pairings shall adhere to the following conventions in their photographic representation:
PP1 will remain fixed in place, and PP2 will move relative to it, to show which edges are being linked.
Each photograph will represent one set of examples showing the four ways in which one edge of PP1 can be paired with the four Length B edges of PP2, rotating clockwise around PP1.
Set 1 illustrates the pairings to one edge of PP1, to the four edges of PP2, when seen Top View to Top View.
Set 2 illustrates the four pairings when seen Top View to Bottom View.
Set 3 illustrates the four pairings when seen Bottom View to Top View.
Set 4 illustrates the four pairings when seen Bottom View to Bottom View.
Step 8: Takeaways
For Length A, we find that for each relationship - Top/Top, Top/Bottom, Bottom/Top, and Bottom/Bottom - the sixteen ways of pairing are the same, and of the sixteen, eight are mirror images of the other eight.
For Length B, it is only four possible pairs that repeat similarly, with two sets of mirror images.
Next up: We'll look at how the various pairs hinge!
Stay safe everyone!
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