Let’s make something!
2D to 3D
loops
Of course, the simplest way to make 3d from 2d is by taking a flat 2d rectangle strip and just connecting its opposite ends with tape or glue. This creates a cylinder (’tube’ or ’loop’) that can stand on its own. With a few cutout shapes, this is great for making paper crowns.
If you connect the opposite ends of a basic loop at the same sides, you get a circular cylinder. But if you connect the opposite ends at the same side, you get more of a ‘drop’ shape.
You can do a lot of the same things with both, but because of their shapes, drops and circles connect to each other in different ways.
If you connect drops on their flat little tab, there’s a limit of about 5 or 6 that you can glue together into a ‘half flower’:
As long as you leave the opposite ends complete, you can trim holes in the center of circles or drops to make them look like many shapes floating on top of each other.
Both circles and drops can be glued together on their rounded sides into larger ‘circular shapes’:
Circles and drops can also be glued together into ‘rows and grids’:
(and then you can glue them together into ‘half flowers’, ‘circular shapes’ or ‘rows and grids’)
‘Circle’ and ‘drop’ loops can also be ‘stretched’ out of a piecer of paper with many slits. (see tutorial #01 for more info about ‘stretched’ 3d shapes) Start by cutting a square piece of paper with slits (use the template at the end of this tutorial that’s called ‘STRETCHED LOOPS’).
Starting from the center, connect opposite ‘half squares’ into circle loops or drop loops. Alternate front and back.
Another thing you can make with circle loops is ‘loop spheres’. If you nest 6 loops inside each other at 30 degrees from each other, you end up with a complete sphere: 30 degrees
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If you add a half-twist before connecting the rectangle strip, the twisted loop is a unique shape called a mobius strip.
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Mobius strips are unique because they have only one face and only one edge. If you draw a continuous line on its face, you’ll see that you go to the front AND back and end up exactly where you started (front and back are the same). And if you draw along one edge you’ll draw top AND bottom and also end up where you started (bottom edge and top edge are the same).
Note: if instead of a half-twist you add two half-twists (a full twist) before connecting the rectangle strip, then it’s no longer a mobius strip. Mobius strips need odd numbers of half-twists. If the number of half-twists is even (2, 4, 6...) then the 3d shape is just a twisted band. Because of their unique shape, something different happens if you cut along the length of mobius strips in two halves or if you cut in ‘thirds’ (or anything less than half).
As you might know, if you cut the length of a simple cylinder along the midpoint, you get 2 narrower cylinders. And the same thing happens if you cut closer to an edge (for example: one third of the width).
But if you cut the length of a half-twist mobius strip along the midpoint, you’ll actually still get a single loop (longer, narrower, and with 4 half-twists).
And if you cut the length of a simple half-twist mobius strip closer to one edge (for example: one third of the width) then you end up with two separate interlocked loops! (one shorter and wider with a half-twist, and another longer and narrower with 4 half-twists)
Twisted bands are similar to mobius strips, but not exactly. If you cut the length of a twisted band along the midpoint, you’ll end up with two separate interlocked loops. But the two will be of equal length and number of half-twists.
With twisted bands, it doesn’t matter if you cut exactly at the midpoint or closer to the edge: you always get two separate interlocked loops of equal length and number of half-twists. (cutting closer to the edge just makes one of the two loops narrower than the other)
mobius strips:
topology:
There are math rules that show what you get when you cut mobius strips and twisted bands):
Topology is the field of math that looks at deformed objects like mobius strips (’deformed’ because they were originally simple cylinders).
If you cut at the MIDPOINT: Mobius Strip: Result is one longer loop (twice as long but half width of original mobius strip and with 2N+2 half-twists) N = original number of half-twists
With topology one can study how shapes are put together (and not just the way they look).
Twisted Band: Result is two interlocked loops: Each is same length and number of half-twists as the original band (but half the width). If you cut at ONE THIRD FROM EDGE: Mobius Strip: Result is two interlocked loops: One is same size as original mobius strip (and same number of half-twists, but 2/3 as wide) The other loop is twice as long (1/3 width of original mobius strip and with 2N+2 half-twists) Twisted Band: Result is same as MIDPOINT, but: One loop is 1/3 width of the original and the other loop is 2/3 width of the original.
So, for example, the shapes of these letter are similar because of how they look:
M, N, W
But all of these letters are similar because of how they are put together:
C, G, I, J, L, M, N, S, U, V, W, Z (these 12 letters are all bent bars with no
holes; compared to one hole and one tail, or
and
which have
and
one bar and four tails)
which have
TWISTED BANDS AND MOBIUS STRIPS cut in half here
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1
2
2 cut closer to edge here (one ’fifth’)
Try these with mobius strips (one half-twist before taping the ends) Try these with twisted band (two half-twists/one full twist) before taping the ends
CROWN 1
note: print this large on tabloid paper (11x17 inches)
front 5
4
5
2
1
4
6
7
8
9
about 15 inches long
4 1
3
optional back extensions (trim length at end #3 to head size)
5 3
5 2
4
CROWN 1 DECORATIONS note: print this large on tabloid paper (11x17 inches), same scale as crown 1
4
5
6
4
5
7
8
4
5
4
5
9
CROWN 2
note: print this large on tabloid paper (11x17 inches)
1
2
front
about 15 inches long
2
1
3
3
optional back extensions (trim length at end #3 to head size)
STRETCHED LOOPS
LOOP SPHERE use this 30-degree grid to align your 6 circle loops