Liked this design in a gif I found for Carnival of Mathematics 108, so I had to play around with the underlying ideas. My first try was dilations, which warps the kites much more quickly. (But is still pretty.) This keeps a linear pattern to the points that determine the kites. The spiral effect seems strongest around t=1.5, but I haven’t thought much yet about why. (The middle image is a gif of the variation of t, click on it if it’s not animated in your dash.)
On GeoGebraTube for playing.
The 108 Eyes gif maker chose 9 layers of 12 kites, which is a nice effect. The last image is 12 layers of 9 kites… much less aesthetically pleasing.
Saw the nice tumblr Spirographie, which suggests a cool cycloid parametrization in it’s subtitle:
x=(R+r)sin(t) + psin(t/r)
y =(R+r)cos(t) + pcos(t/r)
So I made a GeoGebra sketch, of course! I think it’s pretty good for experimenting with the supercycloids and making pretty pictures. Enjoy! Let me know if you (a) make a pretty picture or (b) have ideas for features.
Construction of a Pentagon for Carnival of Math 108.
I’m hosting at my main blog. Tweet (@mathhombre), email (goldenj-at-gvsu.edu) or use the carnival submission form by March 5th. Be nice to see some Tumblr math blogs in it!
A saw that neat math gif at the bottom at mathani on Tumblr
This sketch started because of the “I have to make this” moment, but then I added a 2nd chord in the other circle, and an unconstrained point to compare it with.
So why is the segment of constant length?
On GeoGebraTube for you to investigate.
I made a Fibonacci Spiral GeoGebra sketch that I used to make these. It has two tools to draw arcs in a Fibonacci spiral (so that the radii of two arcs add to the radii of the next). I used those tools to make one image, than rearranged the first three points to make these four images.) You can play with it on GeoGebraTube.
Carlee Hollenbeck wrote that her students were playing with this fun pattern, wondering about the number of triangles. (From Fawn Nguyen’s great site visualpatterns.org, #38.) So I made a GeoGebra sketch to help them. (On GeoGebraTube.)
Here’s what I wrote back:
So I see these in two ways, one just thinking about how many of the smallest triangle, which is not the triangular numbers.
1, 1+3, 1+3+5… 1,4,9 square numbers!
It’s interesting to separate them into right side up and upside down…
RU: 1, 3, 6, 10…
UD: 0, 1, 3, 6,…
But in the broader puzzle sense with how many triangles of each size,
1, 4+1, 9+3+1, 16+__+3+1
The blank is for how many of the 2 size. 6 point up, and 1 point down?
So maybe it’s helpful to separate for each size up and down.
2: 5 total
Digger in GeoGebraI was watching a construction site, wondering about the odd shape of the digger’s main arm. Why would it be bent? Physics and leverage? I asked a physicist and he didn’t know.
Then I saw a chart and realized it was geometry!
Play around with your own design, and compare it to the actual. On GeoGebraTube, of course. Would it be possible to make a digger that reaches too far?
Many Different Ways of Obtaining an Ellipse
In mathematics, an ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve. As such, it is a generalization of a circle which is a special type of an ellipse that has both focal points at the same location. The shape of an ellipse (how ‘elongated’ it is) is represented by its eccentricity which for an ellipse can be any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.
There are many different ways of forming an ellipse. Above are a few examples!
- An animation of the Trammel of Archimides.
- An animation of Van Schooten’s Ellipse.
- An ellipse as a special case of a hypotrochoid.
- Matt Henderson’s animation of a curve surrounding two foci.
Can you think of other ways of forming an ellipse (there’s a really obvious method that isn’t listed above…)?
One of my favorite topics! All of these were inspirational. So here’s the GeoGebra versions: generalized Archimedes’ Trammel, Van Schooten, two circles, matthen’s definitional
Bicorne Graph and Hat
Inspired by a Tumblr link from geometricloci to this page from the curve encyclopedia.
A proper bicorne graph has tangents at the corners that pass through the vertex. Is the actual bicorne hat a mathematical bicorne?
I don’t understand the construction of the false bicorne on this page, though! I can’t determine how the locus point is constructed.
Play with this interesting family of curves yourself at GeoGebraTube.