Irrational Memories

Back when I was young enough to care to try to list such things, I had a favourite number. Really, really faourite. I lived and breathed that number for a while. Today’s session in the freshman seminar “The Art and Science of Seeing and the Seeing and Science of Art”, about which I have blogged here and here, was all about it. Rather than do chapter and verse about it (don’t get me started!), I will instead leave you with the image that I ended with…

penrose tiling

… and let you tell me and other readers – if you like – what you think the number is, what it means to you, and perhaps share whatever you like (or hate) about it.


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16 Responses to Irrational Memories

  1. anonymous says:


  2. anonymous says:

    I mean PI!

  3. Clifford says:

    Good try…. no, it is a real number that I am thinking of. It is embedded in that pattern (and all over art, science, and Nature) in myriad ways.


  4. Dissonant says:

    The Golden Ratio.

  5. Aaron F. says:


    Four? 😉

  6. ioana says:

    three? I see a lot of cubes here 🙂

    is this “living and breathing” the number like counting rocks in the pavement and such?

    ps. Golden ratio is a number????

  7. Navneeth says:

    It is the golden ratio. I think there’s a post/comment somewhere here, or in some other blog, where Clifford mentions it as his favourite number. Of course, a lecture on the “science of art” will not end without mentioning phi. 🙂

  8. Robert says:

    My guess (before seeing the picture) was 4: I like it, it’s not 1, it’s not a prime thus has some structure (being 2×2 and 2 sqared), thus I would consider it the first interesting number. Then I saw the picture and realised: I was off by one, it must be 5! There are 72 degree angles everywhere in the picture and those come from pentagons and pentagrams. You might have the impression that it’s phi, the golden ratio but that is obviously wrong, phi is only derived from 5. In fact, to construct a pentagon with compass and ruler, what you in fact have to construct is phi. Once you have that the rest follows easily. I still remember how proud I was once I had figured that out as a kid.

    I really think phi is far overhyped. In places where it really shows up (in a mathematically provable way) it always comes as the root of the simplest non-trivial polynomial: Non-triviality comes in because this is what makes people notice (nobody would be excited about all the ways 1 or 2 appear in nature and maths). Linear functions don’t have roots which are in any sense non-trivial. Thus we need a quadratic polinomial. the roots (as far as they are real) of X^2 and X^2+1 and X^2-1 and X^2+X and X^2-X are not really exciting and the next thing one woul d try is X^2-X-1 and here you are, phi! Thus phi appears everywhere, where the problem boils down to a quadratic equation which is simple but not too simple.This includes the golden ratio 1/X=X/(X+1) and the Fiobinacci numbers (where the quadratic equation appears once you try to find which geometric series obeys the recurrence relation) and the continued fraction expresseion.

    The appearances in biology (not the rabbits which have a strange sexual behaviour which relate to the Fiobinacci numbers) but the angles of leaves, Leonardo’s ratios of the human body are far too unprecisely defined than to be more specific than “any number between 1 and 3/2”. So you are free to take phi as that number but if you were culturally inclined to see pi/e everywhere I bet you would find it in the same biological systems in the same place as phi.

    One final thing about Penrose tilings (the picture being part of one: non-periodic tilings of the plane with two pieces) I would like to share is the geometry of the space of all these tilings (as I learned from Connes’ book): There are infinitely many different ones, but they only differ at infinity: That is take any finite part of a such a tiling, then it appears somewhere in every other tiling, still the tilings can be globally different. Connes argues that the space of all tilings is naturally described by non-commutative geometry, see his book.

  9. Ambitwistor says:

    It’s certainly the golden ratio, which is the most irrational number and appears in Penrose tilings. I love this number — I did a report on it in the 10th grade for my algebra class.

  10. Clifford says:

    Yes! It is indeed the Golden mean, or Golden ratio, or Golden number. It is irrational:

    golden mean

    It’s wonderful. Here’s a couple of nice properties: Square it, and you get the same number with 1 added. Invert it, and you get it back with 1 subtracted.

    Just for fun…. count up the number of fat rhombi and the number of thin rhombi in the above figure, take the ratio…. you’ll find it rather close to our special number. Take the ratio of the area of an individual fat rhombus to that of a thin one…. same result. There’s a lot more in there… see some of the things Robert said above, and in the links given by Ambitwistor.

    People wonder why the golden mean seems to have such “pleasing proportions”. See a discussion here. Are here’s another of the many articles collecting together aspects of the golden mean’s appearance in art, architecture, music, science…..


  11. Yvette says:

    The Golden Ratio is a good choice for a favorite number. 🙂

    I had one back in the day. It was five, because when I turned five I decreed it to be so. I was really upset when time marched on and I turned six…

  12. Melquiades says:

    This is not my favorite number, but indeed it sounds strange…

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