# The Mathematics of Logos

As you may know, we created a new logo for Key using a circle and rotational animation to create a visual representation of pi. Karen Coe describes this in her post Branding Isn’t About Math. Are You Sure About That?, which also includes a video of how the logo was created.

In the article, Karen talks about how using math and our own software, Sketchpad, to create our new logo was a great way to walk the talk – we sell math software, promote and support quality math education, so our logo is math. It’s always a great thing when math can be used in a relevant, real-world context.

So I was excited when a colleague forwarded a story about Twitter’s logo design update that talked about how three sets of three interlocking circles were used to create the new, streamlined Twitter bird. Another example of a logo created using math. (Be sure to watch their quick video clip of it.)

Naturally, I had to play around on Sketchpad to see if I could recreate the Twitter bird.

I haven’t quite figured it out yet, but that is part of the fun, isn’t it? The challenge of trying to determine what went on behind the scenes. How does this work? What size circles did they use? What parts were colored from the intersections? For those of you out there, give it a try yourself or let your students try—it requires a bit of understanding of arcs, arc sectors, intersecting circles, and so on.

Of course, all of this led me on a math-logo treasure hunt. What other logos have mathematical constructs behind them? Obviously we could say many, as most logos are geometric in nature.  But it’s not always obvious what the underlying math construct is, so wouldn’t it be interesting to try and figure it out, using a dynamic tool like Sketchpad to try to recreate some logos and discover the mathematical constructs and relationships? Perhaps a fun summer project for students, or a potential project to explore for next year.

I found this great website, Famous Logos, where you can click on a logo of interest and it tells you about how the logo was developed, what it represents, and its design elements. So, for example, the Volcom logo looks like an inverted diamond, but it’s really a polygon with various triangles reflected over a line (see my Sketchpad version to the left). My treasure hunt was purely geometric in interest—what could I try to recreate with Sketchpad? Therefore, most logos with words in them I wasn’t interested in. Here are few that are interesting in relation to this treasure hunt.  How would you reconstruct them? What’s the mathematics behind them?

1. The Girl Scout Logo which includes translations of images
2. The Toyota Logowhich involves intersecting ellipses
3. The Steelers Logo which contains three ‘asteroids’ (I thought they could be created using line designs on a coordinate axis) within a circle
4. The Batman Logo which definitely involves reflection
5. Chevrolet Logo where we have possibly parallelograms, rectangles and reflection or translation?  Probably a couple ways to recreate this logo.
6. Mitsubishi Logo which involves the rotation of a rhombus

Explore for yourself and let your students explore as well. Great creativity lessons and a nice answer to the ever-present question, “When are we ever going to use this?” Clearly—math and logos go together!

I have been in the math education field for over 20 years. The first 15 of those as a middle and high school math teacher, 2 of those years as a district math specialist, and the last several years working as a professional development provider. I am incredibly interested and motivated by technology in education, for teachers and students, and ways to incorporate technology effectively into mathematics so that students experience and explore math in dynamic ways. I am continually trying to learn about how to improve teacher instruction so that students can learn and understand mathematics better. I have a math degree from Virginia Tech (Go Hokies!!), a Masters in Curriculum and Instruction (Math focus) from Virginia Commonwealth University and am currently ABD (all but dissertation) in Educational Technology from the College of William and Mary. I currently work for Key Curriculum Press as their Director of Education Technology Outreach - a dream job, working for the company that helped me through years of teaching...I have to pinch myself most days!
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### 5 Responses to The Mathematics of Logos

1. Shawn Urban says:

Another math-created logo is the Nike swoosh, which is a Bezier curve based on the relation of rectanglular perimeter to width.

• Thanks Shawn! That logo is on the FamousLogos site as well, so hopefully folks will explore it. I appreciate you identifying the name of the curve – did not know that!!

2. This is a nice project to give students; I think many students don’t see math as an artistic tool. Many of these logos (and the Twitter one in particular) make use of lunes. (A lune is the region between two intersecting circular arcs; its name is derived from the Latin word for moon, because it looks like a crescent moon.) Because lunes are not built into Sketchpad, these regions have to be constructed. I’ve just posted on Sketch Exchange a sketch that sets out the construction challenge (with some hints) on page 1, shows the completed construction on page 2, and shows a part of the Twitter logo construction on page 3. You can find it here on Sketch Exchange:
http://sketchexchange.keypress.com/sketch/view/510/constructing-a-lune

• Wow – thanks for the lune information and also the cool Sketch. I am excited to start playing with it!!

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