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Archive of posts filed under the Art category.

Using Mathematics to Create — Geogebra

Geogebra Student Work -- TriangleOne of my guiding principles as a math teacher, as I articulate in this TEDx talk, is to provide students with tools and opportunities to create with mathematics.  Few things are as aligned with that principle as well as Geogebra, the free, open-source, dynamic geometry environment.

I’ve integrated a lot of Geogebra work in my Geometry class this year.  I use Geogebra assignments to assess basic geometric skills, to connect old ideas to new, and to explore geometry dynamically.

But much like geometry itself, once you master a few elementary rules in Geogebra, you can create amazing and beautiful works of mathematics.

Below is an example of some wonderful student work from this year.  After an introduction to polygons, students were given two simple ideas for creating new objects from polygons:  constructing diagonals and extending sides.  I gave students some technical tips on how to color and polish their final products, and invited them to be creative.  As usual, they did not disappoint.

Geogebra Student Work -- Combination

Students, and teachers, need more opportunities to create with mathematics.  We’re fortunate to have technologies like Geogebra that offer us those opportunities.

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Happy 2016!

In honor of the new year, here’s the complete graph on 64 vertices, with its 2016 edges!


complete graph is a graph in which every pair of vertices is connected with an edge.  In a complete graph with n vertices, there are

\binom{n}{2} = \frac{n(n-1)}{2}

edges.  The above graph has 64 vertices equally spaced around the perimeter.  Thus, n = 64, and we have

\binom{64}{2} = \frac{64*63}{2} = 2016


The number 2016 is special for a variety of reasons.  For example,

1 + 2 + 3 + ... + 63 = \sum\limits_{n=1}^{63} n = 2016

So 2016 is equal to the sum of the first 63 positive integers!  This makes 2016 a triangular number, a fact beautifully demonstrated by David Swart in this image.

And John D. Cook illustrates the combinatorial nature of 2016 by pointing out that this is the number of ways to place two pawns on a chessboard!

However you think of it, 2016 is a pretty great number!  And here’s hoping 2016 is a great year.

Math Photo: Slope Field in the Sky

Slope Field in the Sky

I finally made it down to the new Fulton St. subway station.  Such a beautiful view from below, like the slope field of a differential equation.

Each of the 88 glass blades of the Sky Reflector-Net is positioned to channel sunlight down throughout the station at different times of day and days of the year.  Maybe next year I’ll stop by on the summer solstice.

Bridges Math and Art 2015

Bridges 2015I am excited to once again be participating in the Bridges Math and Art conference this summer!

The Bridges organization has been hosting this international conference highlighting the connections between art, mathematics, and computer science since 1994.

I have participated in several Bridges conferences, and my experiences there have greatly influenced me as a mathematician and a teacher.

This year, I’ll be presenting a short paper, “Monte Carlo Art Using Scratch“, chairing a short paper session, and exhibiting a photograph in the Bridges Mathematical Art Gallery.

You can view the Bridges 2015 program here, see the entire 2015 Art Exhibit here, and learn more about the conference and the Bridges organization here.

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Math Photo: Bond Angles

Bond Angles

I love the way the struts of this structure come together and align with the geometry of the city.