Here are some examples of 3D surfaces sculpted out of floral foam by a Calculus student.
Creative and beautiful! More examples can be seen on my Facebook page.
Application Geometry
Geometry of Pasta
This is a fun slideshow of images from architect George Legendre’s book “Pasta by Design”
http://tmagazine.blogs.nytimes.com/2011/09/27/using-his-noodle/
The slideshow, from the on-line New York Times Style magazine, shows a few different renderings of pasta shapes. The book, apparently, contains actual cooking directions as well as these technical schematics.
It appears as though Legendre used a graphing utility like Mathematica or Maple to sketch out his pastas. A creative, and fun, application of this technology!
Challenge Geometry
Which Triangle is More Equilateral? Part II
A surprising amount of interest was generated by my question “Which Triangle is More Equilateral?” With the passing of consecutive Isosceles Triangle days, I wondered: which triangle was more equilateral, the 10-10-11 triangle, or the 10-11-11 triangle?
Many rich and interesting conversations arose as a result of this question. Colleagues, students, and commenters on the original post offered good ideas about how to approach both the question itself and the concept in general.
There is a lot to say about this seemingly simple problem, but I’ll begin by sharing my approach to the question “Which triangle is more equilateral?”
After playing around with several ideas, I tried to get to the heart of equilateralness. Equilateral is a well-defined idea (all sides congruent), but how could we relax that definition and quanitfy equilateralness in a continuous, rather than a discrete, way?
I chose to to think of equilateralness as a measure of how circle-like the object is: the more it’s like a circle, the more equilateral that object is. What does it mean to be circle-like? I chose the following idea as my foundation: the fundamental characteristic of the circle is that it maximizes area for a given perimeter. From that perspective, I created a measurement.
I define a triangle’s equilateralness to be the ratio of its area to the area of the equilateral triangle with the same perimeter. Thus, for triangle ABC with sides a, b, and c, its equilateralness is given by
where [ABC] denotes the area of triangle ABC.
Since the equlateral triangle is the triangle of maximum area for a given perimeter, the above measure will be bewtween 0 and 1 for all triangles. The closer its Eq is to 1 , the closer the triangle is to being equilateral.
So for the triangles in question, we have
Thus, by a very slim margin, the 10-11-11 triangle is more equilateral than the 10-10-11 triangle!
The value in this question is not so much settling on this, or any, particular approach; what’s valuable here is the opportunity to creatively explore a lot of interesting and deep mathematical ideas. I look forward to doing just that!
Related Posts
- Which Triangle is More Equilateral?
- Another Equilateral Comparison
- Which Triangle is More Equilateral? 2012 Edition
- An Ode to Equilateralism
Challenge Geometry
Which Triangle is More Equilateral?
Two consecutive isosceles triangle days (10/10/11 and 10/11/11) got me thinking: Which of the following triangles is more equilateral, the 10-10-11 triangle, or the 10-11-11 triangle?
Or perhaps I should say equilateraler?
[Update: here is my first attempt to answer this question in a mathematically sensible way.]
Related Posts
Appreciation Geometry Uncategorized
Dodecahedron Calendar
Here’s a fun use for a dodecahedron: folding it up to make a yearly calendar!
https://texample.net/tikz/examples/foldable-dodecahedron-with-calendar/
Just download the PDF, print, cut, fold, and glue! Access to a large-scale plotter might be nice, as the 8.5 x 11 version folds into something that’s pretty small.
It’s too bad there aren’t eight days in a week, otherwise we could put the octahedron to use, too!