Self-similarity in math & ambigrams 3/3

by | Friday, August 08, 2014

Self-similarity in geometry is the idea of repeating a similar shape (often at a different scale) over and over again. In other words, a self-similar image contains copies of itself at smaller and smaller scales, such as the image below of the word “zoom.”


Self-similarity is a rich mathematical idea and connects to other powerful concepts such as infinity, iteration, fractals, recursion and so on. As it turns out self-similarity is also a rich source of ambigrams. This is why the third article in the series Of Art & Math is devoted to Self-Similarity. This series written with my friend Gaurav Bhatnagar is published by At Right Angles (a mathematics education magazine). 


I think this is easily the best article of the three we have written so far. It has some of the best original designs I have created. Gaurav pushed me hard mathematically, and I dare say, I met the challenge (at least part of the way). I don’t want to reveal too many of the designs in the article (links to download the article are given below) but here are a couple. Below are two different designs for the word “Infinity.”


These two different ambigram designs for “infinity” are subtly different from each other. In both cases the word can be read even when you rotate the design around – both at the top of the circle and the bottom! Notice how in the first design the chain is created by “in” mapping to itself and “finity” mapping to itself. In contrast the second design breaks the word up differently, mapping “ity” to “in” and “fin” to itself. In addition the first design wraps around a circle – for ever and ever and the second says infinity both in words and in symbol!

The idea of infinity is captured somewhat differently in the next two images.

infinity-2 styles

The first focuses on mapping the design onto a sphere while the second is a self-similar shape that circles inwards forever. In either case the design can be interpreted in two different ways. Either being made of an infinite repetition of the word “finite” or the infinite repetition of the word “infinite” (where the shape that reads as the last “e” in the word “finite” can be read as “in” in the word “infinite” when rotated by 90 degrees).

There are lot more designs in the actual article. If you love math or ambigrams are just interested in exploring some cool ideas, go ahead click the links at the end of this post.

All in all this series seeks to reveal the hidden beauty of mathematics – and thus it is only fitting that it ends with this design for “hidden beauty.”


You can download each of the articles in the series Of Art & Math by following the links below

  1. Introducing Ambigrams: Blog postDirect link to PDF
  2. Symmetry: Blog post | direct link to PDF
  3. Self-SimilarityDirect link to PDF

Alternatively you can download all three articles in one large(-ish) PDF by clicking here.

A few randomly selected blog posts…

Create… Emergence!

I ended the mashup TPACK commercial with the following words, Explore, Create, Share! Over the past couple of weeks my kids and I have been working on a series of short videos around these three words. The first one we made was around the idea of "Explore," titled To...

Words in 3 Dimensions

Words in 3 Dimensions

A few weeks ago I started doodling words in 3 dimensions, for no particular reason, and before I knew it I had a bunch of interesting designs. Here is a sample: A bit of goofing around with Keynote and some royalty free music from Kevin McLeod, and I had a little...

21st century learning, TPACK and other fun stuff

I have been invited to participate in the 2014 Educational Technology Summit: Empowering Educators to Enhance Student Learning in the Digital Era. This conference is being organized by Common Sense Media, Annenberg Retreat at Sunnylands, & the LEAD Commission. I...

Sketching on the iPad

Over the past few weeks I have been experimenting with using my iPad as a drawing/painting tool. The sketches below were created by tracing on an existing image - usually a photograph. So this is not "freehand" drawing per se - but given my limited talents that may...

21st Century Learning, one school’s ongoing story

Recently I had been invited to the Birmingham School District to speak to the administrators, teachers and broader community about their recent initiatives on 21st Century Learning. I had a wonderful visit - which I was reminded of by this article (On the Front Lines...

STEM Futures at AAAS

STEM Futures at AAAS

ASU recently hosted, what is known as, the world's largest scientific gathering, the annual conference of the American Association of the Advancement of Science. As as part of this conference I was invited, along with Ariel Anbar and Trina Davis, to talk about our...

Research conduct: The movie

From Ken Friedman & the PhD Design listserv: The current issue of The Scientist has a story on an interactive film that helps research students and early career researchers to understand and navigate the perils of research misconduct. Highlights: "The Lab is a...

Charleston, SC for SITE 09

I am off to Charleston, SC for the SITE 2009 conference. . I can't believe it has been a year since Matt Koehler and I presented our Keynote. I am sending this note sitting in the Michigan Flyer bus (making good use of their free wi-fi) and am looking forward to a...

Obtuse can be right!

My daughter, whose creative exploits have been featured here before (for instance see her design for a math-music game), now has a blog, titled Uniquely Mine. It features original writing (poems, stories) by her. Do check it out. You can find regular updates on this...

1 Comment

  1. cool math games

    cool post



  1. Infinite Regress: New ambigram / visual pun – Punya Mishra's Web - […] regress.” I have created many ambigrams to represent the idea of infinity (click here for examples) but this one is different…

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