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Daniel Piker Profile picture
@KangarooPhysics
Sep 17, 2021 • 7 tweets • 3 min read • Read on X
Scrolly
To give a bit of explanation of this and some of the knot surface animations I've posted previously-
The solid angle of a curve is the signed area inside its projection onto a unit sphere (note the flip as it moves around the outside).
We can consider this value at all points in space as a scalar field and generate it for any closed oriented curve.
Taking the level set of this field we get a mesh which spans the curve.
Varying the value we take this contour at we can get a whole family of surfaces filling space.



To combine multiple curves we can simply sum the values.

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More from @KangarooPhysics

Daniel Piker Profile picture
Feb 14, 2023
The optimal known packing of 17 equal squares into a larger square - i.e. the arrangement which minimises the size of the large square. Image
(I saw a copy my old post was getting picked up again, and thought people might like a higher resolution image of this beauty.) Generated from the data in link.springer.com/article/10.100…
I picked the solution for 17 because that one looked particularly funny to me, but it's not just this number with an awkward looking solution - the arrangements for most numbers are asymmetrical, the neat ones like 16 are the exception.
erich-friedman.github.io/packing/squins…
Read 7 tweets
Daniel Piker Profile picture
Aug 17, 2022
Here's a fun little kinematic toy that's easy to make - an equilateral orthogonal hexagon linkage with 1 degree of freedom:
Just take 6 squares of stiff cardboard and tape them together along half the length of their edges. You can start with them flat and work your way round in a spiral. To make it a bit stronger, you can put tape on both sides of each hinge. Image
Once you get to the last one, to close the loop you need to arrange them in 3d so the inner opening forms a shape like a seat (if you join them in the other arrangement it will be rigid and not a mechanism).
To tape this last pair, it can help to place them on the edge of a table Image
Read 6 tweets
Daniel Piker Profile picture
Dec 9, 2021
The best known packings of N equal circles into a square and a circle.
Best in the sense that they are the arrangements which maximise the circle radius without overlaps.
Patterns don't always hold, and the most symmetrical solution isn't always best.
Coordinate data and references for all these can be found at packomania.com
I made a little puzzle that locally optimizes for these while you interact to find the arrangement that matches the known global optimum.
Read 6 tweets

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