Some time ago, Bodo posted a transition unit challenge in Origami Design School Discord server. I solved them, and thanks to that, my understanding of the construction improved. I'll share the screenshot of the challenge so you can try solving it.
Problem Statement
Complete the following crease patterns so they are flat foldable!
(credit to Bodo)
Note: bottom left crease has a missing crease, you are free to add the extra crease in the left border.
Observation
The first row has a similarity: both of the left and right border has a 1 unit pleat, while the top and bottom has a single mountain fold.
The second row is also similar, but the left side has 2 unit pleat, and the right side has 2 one unit pleats. This is equivalent to transition a 2 unit pleat into 2 one unit pleats.
Finally the third row has 3 unit pleat to be transitioned into several pleats whose sum is 3.
In fact, it is always possible to transition an N unit pleat into multiple smaller pleats whose unit size sum to N, with arbitrary gaps separating each of them. See this post for example of transitioning 1 unit pleats into 1/8.
Construction
The transition is built constructively, so it's not drawn in big bang.
We can start with the simple diamond shaped squash.
Say that we want to transition it to 2 half unit pleats, separated by gap.
Start by making squash on top of the squash all the way to where the topmost of the pleat's crease is.
The purpose is to make a sloped crease highlighted in yellow.
Note that this squash can be any size and be skewed in any direction. We will start with normal symmetric squash that divides the diamond's top by half, and shall see later what happened if different choices are made.
With this sloped crease, we can stretch the squash to fit the topmost pleat. First connect the topmost crease with the sloped crease (orange), erase the creases in between, then find the 4th crease that will hit the 2nd topmost crease (yellow). The remaining creases will connect nicely to complete the transition.
On real paper this is quite fun to do and straightforward once you have all the vertical/horizontal creases precreased.
Now let's try different choices of squash. The same construction can produce valid transition with symmetric squashes of various sizes
The transition size gets wider when the initial squash is skinny. Note that the leftmost one has the squash eat the entire top half of the diamond, cancelling the cyan crease.
Can we make the squash taller than necessary? Yes but it will be wasteful and arguably harder to fold in real paper. So there's no apparent benefit.
We can also try asymmetric squash, which turns out to produce the same result as if the entire top half of diamond was eaten by the squash.
This construction is flexible. We can even transition the 1 unit pleat to 1/4 and 3/4 pleats separated by gap. Choose whichever squash you prefer.
 |
| Split into 1/4 and 3/4 |
 |
| Split into 3/4 and 1/4 |
The idea of using squash to transition can be repeated. Say that we want to transition 1 unit pleat to 1/4, 1/4, and 1/2. First do the topmost pleat, then apply the same construction to create the 2nd from the top. Again we can choose any kind of squash for this 2nd transition. The final result is a compound transition unit. Follow the steps below from top to bottom (last 2 steps have 3 different paths)
Sample Solution
Below is my solution as reference, even though some of them are not the best solution. It is purposely posted small but you can click it to see the full resolution.
Highly recommended to try solving it yourself.
Comments
Post a Comment