Slope Demonstrations
While working with drawing crease patterns on square grids, it's useful to know what slopes can be used in combinations to lie flat. For example, if you have 3 of the 4 necessary creases that meet at an intersection, the 4th one MUST be at a forced angle, but sometimes determining that angle would not be so easy.
These are some sample slope "calculations" that I have used in the past and find quite useful in combination with Kawasaki's Theorem.
In all 4 diagrams, the 90 degrees in the lower left corner is broken up into 3 angle-segments. The "equations" below are all shown in the form that the lowest segment plus the middle segment equals their sum.
Top Left: 1/3 + 1/2 = 1
Left Middle: 1/2 + 1/2 = 4/3
Top Right: 1/3 + 1/3 = 3/4
Bottom: 1/3 + 1/7 = 1/2
The first slope in each equation is straight forward, as it's just the slope of the lower line from the lower left corner.
The third slope (to the right of the equals sign) is also straight forward, as it's just the slope of the upper line from the lower left corner.
The 2nd slope is the only tricky one. Each diagram has a right triangle shown, with the right-angle marked. The proportion of the lengths of the two legs of right triangle (the edges joining at the right angle) gives us the 2nd slope.
Let me know if it isn't quite clear enough what I mean here and I will try to clarify as needed.
Note: One interesting thing is that all of the angle bisectors of a 3-4-5 triangle will intersect at a point on the grid. This has been used in the past for a few representational models. I believe Satoshi's Dragonfly is one example.
Slope Demonstrations
While working with drawing crease patterns on square grids, it's useful to know what slopes can be used in combinations to lie flat. For example, if you have 3 of the 4 necessary creases that meet at an intersection, the 4th one MUST be at a forced angle, but sometimes determining that angle would not be so easy.
These are some sample slope "calculations" that I have used in the past and find quite useful in combination with Kawasaki's Theorem.
In all 4 diagrams, the 90 degrees in the lower left corner is broken up into 3 angle-segments. The "equations" below are all shown in the form that the lowest segment plus the middle segment equals their sum.
Top Left: 1/3 + 1/2 = 1
Left Middle: 1/2 + 1/2 = 4/3
Top Right: 1/3 + 1/3 = 3/4
Bottom: 1/3 + 1/7 = 1/2
The first slope in each equation is straight forward, as it's just the slope of the lower line from the lower left corner.
The third slope (to the right of the equals sign) is also straight forward, as it's just the slope of the upper line from the lower left corner.
The 2nd slope is the only tricky one. Each diagram has a right triangle shown, with the right-angle marked. The proportion of the lengths of the two legs of right triangle (the edges joining at the right angle) gives us the 2nd slope.
Let me know if it isn't quite clear enough what I mean here and I will try to clarify as needed.
Note: One interesting thing is that all of the angle bisectors of a 3-4-5 triangle will intersect at a point on the grid. This has been used in the past for a few representational models. I believe Satoshi's Dragonfly is one example.