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Her er en tekst jeg skrev i 2010 for Argument.

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Here's a picture of the finite projective plane over the field with three elements, made with TikZ.

The numbers are the homogeneous coordinates on PP(2,3). Notice how the sum of any two points on the same line gives another point.

The numbers are the homogeneous coordinates on PP(2,3). Notice how the sum of any two points on the same line gives another point.

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Kuuule bilder.

**The graphic nature of groups**

Group theory is an area of abstract algebra that can be used to produce visually appealing pictures like these ones. The underlying algebraic structure here is the set of all sequences of length

*d*whose entries are integers in the range 0 to

*n-1*. Two such sequences can be added together entry by entry, like vectors, where all the additions are computed modulo

*n*. This set, equipped with this operation, is known as the

**group**

*(Z/nZ)^d*. Each of the pictures is generated by choosing a value of

*n*, a value of

*d*, and one sequence from the group.

Groups measure symmetry in the same way that numbers measure quantity. Furthermore, groups have symmetries of their own, known as

*automorphisms*. For example, if we scramble up the

*d*entries in one of the sequences of

*(Z/nZ)^d*using some fixed permutation, this gives an automorphism of the group. (The key property here is that adding two sequences and scrambling the result gives the same answer as scrambling each sequence in advance and then subsequently adding them.) To generate each picture, we scramble up its defining sequence using every possible permutation of

*d*objects; this produces a collection of sequences called an

**orbit**.

Also associated to the group

*G = (Z/nZ)^d*are the

**irreducible characters**of

*G*. These are all the ways to label the elements of

*G*by complex numbers in such a way that addition in

*G*corresponds to multiplication of the corresponding complex numbers. It turns out that there is a natural one to one correspondence between the sequences in

*G*and these irreducible characters. We can then add up all the characters corresponding to the orbit in the previous paragraph to obtain a

**supercharacter**, which is a function from

*G*to the complex numbers. The pictures are the result of plotting all the supercharacter values corresponding to a particular orbit.

The recent paper http://www.arxiv.org/abs/1305.4913 by Brumbaugh, Bulkow, Garcia, Garcia, Michal and Turner contains an astonishingly diverse set of pictures like these ones. The parameters given in each caption give complete information on how to generate the picture. The authors are motivated partly by the aesthetic quality of the pictures, but also by supercharacter theory, which is a generalization of character theory for groups. The paper also explains how their constructions are closely related to Gaussian periods, Ramanujan sums, Kloosterman sums and Heilbronn sums; these are all well-known exponential sums that appear in number theory.

The authors think that the third picture in the series (with

*n*= 173 and

*d*= 3) looks like a manta ray followed by a trail of bubbles. They are also aware that the first picture in the series (with

*n*= 10 and

*d*= 8) looks very creepy, although they don't say so in the paper.

#mathematics

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2013-05-22

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Wow.

**My two-week review of Google Glass: it all depends on the price**

This week I gave five speeches while wearing it.

I passed through airports four times (two more in a couple of hours).

I let hundreds of people try my Google Glass.

I have barely taken it off since getting it other than to sleep.

Here's my review after having Google Glass for two weeks:

1. I will never live a day of my life from now on without it (or a competitor). It's that significant.

2. The success of this totally depends on price. Each audience I asked at the end of my presentations "who would buy this?" As the price got down to $200 literally every hand went up. At $500 a few hands went up. This was consistent, whether talking with students, or more mainstream, older audiences.

3. Nearly everyone had an emotional outburst of "wow" or "amazing" or "that's crazy" or "stunning."

4. At NextWeb 50 people surrounded me and wouldn't let me leave until they had a chance at trying them. I haven't seen that kind of product angst at a conference for a while. This happened to me all week long, it is just crazy.

5. Most of the privacy concerns I had before coming to Germany just didn't show up. I was shocked by how few negative reactions I got (only one, where an audience member said he wouldn't talk to me with them on). Funny, someone asked me to try them in a bathroom (I had them aimed up at that time and refused).

6. There is a total generational gap that I found. The older people said they would use them, probably, but were far more skeptical, or, at minimum, less passionate about the fact that these are the future, than the 13-21-year-olds I met.

So, let's cover the price, first of all. I bet that +Larry Page is considering two price points: something around $500, which would be very profitable. Or $200, which is about what the bill of materials costs. When you tear apart the glasses, like someone else did (I posted that to my Flipboard "Glasshole" magazine) you see a bunch of parts that aren't expensive. This has been designed for mass production. In other words, millions of units. The only way Google will get there is to price them under $300.

I wouldn't be shocked if Larry went very aggressive and priced them at $200. Why would Google do this?

Easy: I'm now extremely addicted to Google services. My photos and videos automatically upload to Google+. Adding other services will soon be possible (I just got a Twitter photo app that is being developed by a third party) but turning on automatic uploads to other services will kill my batteries on both my phone and my glasses (which doesn't have much battery life anyway). So, I'm going to be resistant to adding Facebook, LinkedIn, Twitter, Evernote, and Tumblr to my glasses. Especially when Google+ works darn well and is the default.

Also, Google is forbidding advertising in apps. This is a HUGE shift for Google's business model. I believe Larry Page is moving Google from an advertising-based company to a commerce based company.

The first thing I tried that it failed on was "find me a Sushi restaurant." I'm sure that will get fixed soon and, Google could collect a micropayment anytime I complete a transaction like reserving a seat at a restaurant, or getting a book delivered to my house, or, telling something like Bloomingdales "get me these jeans."

There is literally billions of dollars to be made with this new commerce-based system, rather than force us to sit and look at ads, the way Facebook and tons of other services do.

When you wear these glasses for two weeks you get the affordance is totally different and that having these on opens you up to a new commerce world. Why?

1. They are much more social than looking at a cell phone. Why? I don't need to look away from you to use Google, or get directions, or do other things.

2. The voice works and works with nearly every one and in every situation. It's the first product that literally everyone could use it with voice. It's actually quite amazing, even though I know that the magic is that it expects to hear only a small number of things. "OK Glass, Take a Picture" works. "OK Glass, Take a Photo" doesn't. The Glass is forcing your voice commands to be a certain set of commands and no others will be considered. This makes accuracy crazy high, even if you have an accent.

I continue to be amazed with the camera. It totally changes photography and video. Why? I can capture moments. I counted how many seconds it takes to get my smartphone out of my pocket, open it up, find the camera app, wait for it to load, and then take a photo. Six to 12 seconds. With Google Glass? Less than one second. Every time. And I can use it without having hands free, like if I'm carrying groceries in from the car and my kids are doing something cute.

I've been telling people that this reminds me of the Apple II, which I unboxed with my dad back in 1977. It was expensive. It didn't do much. But I knew my life had changed in a big way and would just get better and better. Already this week I've gotten a new RSS app, the New York Times App, and a Twitter app. With many more on the way.

This is the most interesting new product since the iPhone and I don't say that lightly.

Yeah, we could say the camera isn't good in low light. We could say it doesn't have enough utility. It looks dorky. It freaks some people out (it's new, that will go away once they are in the market).

But I don't care. This has changed my life. I will never live a day without it on.

It is that significant.

Now, Larry, find a way to make it $200 and you'll have a major hit on your hands.

(Attached are dozens of photos I shot over the past two weeks with it).

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April 27, 2013

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Jeg har skrevet en bloggpost om fire farger-teoremet.

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Hva er det med Adeles musikk som gjør den så bra? Svar: Dopamin!

Harmonic analysis on the adeles.

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En altfor vanskelig måte å løse en enkel ligning på. http://cube.fredrikmeyer.net/?p=194

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