This is a generalization of a diagram I saw from d3lt4 on Tumblr:
It is constructed by rotating and dilating the L shape to fit in with itself, rotated 90 degrees. There are some interesting properties that develop based on the height/width ratio and the thickness of the L-shape. At first I thought D3lt4’s diagram was based on an L from a square with thickness half of the width, but instead it is Golden.
The last image is close to real Golden ratio proportions.
On GeoGebraTube for play.
More Poly-Polygon Portals.
So let’s say a polygon has a spiral for a point on it’s border if there’s a center so that a rotation + dilation takes a vertex to that point and each other vertex to a point on a corresponding edge. (Then the next dilation + rotation will have the same property, etc.) Squares have spirals, and rectangles. Any others? Most quadrilaterals seem to have points that do not have spirals, but do most quads have points that do have spirals? No idea. Yet.
Playing around, it seems to depend on the point on the boundary, and I’m guessing the answer is not every polygon has a spiral. For the ones that do, the location of the center seems interesting.
If anyone wants the GeoGebra for this, let me know and I’ll upload. Interested?
Playing around with the Malay Icon shape.
Not sure how this image got into my head. Kept adding features to get what I wanted. In addition to relative size, angles, apothems and the like for math, seems like a good exploration of positive and negative space for art.
On GeoGebraTube for you to play.
A just for fun construction, but pretty to me.
Change the spiral by controlling the angle and distance at the center, or by entering the data you want in the input boxes. The Golden button changes the scale to make the ratio fit the Golden Spiral for the angle you’ve chosen.
The gifs show controlling number, angle and scale. The last still is a Golden Spiral.
On GeoGebraTube, of course.
Here’s the GeoGebra.
EDIT: the function to make the time work was a nice application of the floor(x) function.
Inspired by twitter conversations with Andrew Shauver. This sketch views a matrix to see its effect on a point in the plane by matrix multiplication.
Simple, but surprisingly fun.
Can you make the drawing/writing reverse direction? Flip upside down or left to right? Lean left or right?
What effect do different determinants have?
On GeoGebraTube. Have fun!
This sketch is inspired by a great story at Pat Bellew’s math history blog.
The theorem (illustrated here for the smallest case) involves Esther Klein, Paul Erdos and George Szekeres, plus the birth of combinatorial geometry,
and the marriage of two mathematicians!
This theorem is also covered at Theorem of the Day. (pdf)
The question is: given 5 points in the plane, can you arrange them so any 4 of them form a concave quadrilateral?
Playing around, you may start to wonder if you can make just 1, 3 or 5 concave quadrilaterals!
This requires play; on GeoGebraTube. This is a rare one that works in Java but not HTML5. Best option: download.