Archive of posts filed under the Representation category.

## Proof Without Words: Two Dimensional Geometric Series

I offer this visual proof of the following amazing infinite sum

$\frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{4}{32} + \frac{5}{64} + ... = 1$

## Birthday Frequency Visualization

This is a beautiful visualization of birthday frequency:

http://gizmodo.com/5910773/how-common-is-your-birthday

This “heat map” shows which days are the most common birthdays in the U.S.

Lost of interesting questions arise from this representation of data.  We can immediately see that July, August, and September seem to form a disproportionate band of birthdays.  And take a look at July 4th:  what’s the explanation for that?

In addition, you could also use this chart to create some new twists on the classic birthday paradox!

## Averia: The Average Font

This is a clever and interesting idea:  creating a new font by taking the average of all existing fonts.

http://iotic.com/averia/

By overlaying all the small letter a‘s, say, from all the different fonts, one can take a visual average and create a new letter a.  Repeat for the whole alphabet, numerals, and punctuation marks, and voila!, you’ve got Averia.

The idea of taking a visual average may be a bit mysterious, but the author describes a few different approaches in how to combine the images.  Essentially all of the instances of a particular symbol are placed on top of each other, and the the darkest parts of the new image are where the instances intersect the most.  The result is then smoothed over to create a readable letter.

And the font looks pretty nice, if not too exciting.  Just what you might expect from the average font.

## Combinatorial Bracelets

This is another wonderful visual demonstration from Jason Davies:  a combinatorial bracelet generator.

http://www.jasondavies.com/necklaces/

Combinatorics is the mathematics of counting things, and one of the classic “advanced” counting problems is this:  given a certain number of beads of various colors, how many different bracelets can you make?

The problem may seem easy enough, but it becomes quite difficult when you start to understand what “different” really means.

For example, if you turn one bracelet into another by rotating it, then those two bracelets aren’t different.  Even more complicating is that if you can obtain one bracelet from another by flipping it over, then they are also the same!

This visualization can really help develop a sense of the complicated symmetries at work here.

Click here to see more in Representation.

www.MrHonner.com

## Wind Art

This sculpture by Charles Sowers functions simultaneously as a stimulating piece of art, a representation of data, and an illustration of vector fields.

http://goo.gl/WoXu3

Windswept is a giant billboard covered in little aluminum arrows that twist and turn in the wind.  The arrows can be thought of as bits of data, namely, the direction of the wind that point.  Taken together, and dynamically, they give a sense of the complicated nature of swirling and changing winds.

Just like this wonderful wind map, this sculpture represents mathematical ideas in a beautiful and thought-provoking way!

Click here to see more in Art.

www.MrHonner.com