Portraits of Wildflowers

Perspectives on Nature Photography

Bug nymph on four-nerve daisy

with 36 comments

In contrast to the willful four-nerve daisy flower head (Tetraneuris linearifolia) you saw last time, the flatness of this one that I found on the same April 1st outing had me aiming straight down at it.

You’ll remember that each “petal” of a daisy is actually an independent flower known as a ray flower. The rays (14 in this case) ray-diate out from the flower head’s center, which is made up of many smaller individual flowers of a different type, known as disk flowers. It’s common in daisies for the disk flowers to form overlapping spirals, some of which go out from the center in a clockwise sense, and others in a counter-clockwise sense. If you count the number of disk-flower spirals in each direction, you typically get consecutive Fibonacci numbers. There’s a confirmation of that in the following enlargements of this four-nerve daisy’s disk. Go ahead, count the number of spirals going each way and you’ll see:

In the unlikely event that anyone ever asks you if daisies know how to count, you can confidently and Fibonaccily say yes.

© 2018 Steven Schwartzman

Written by Steve Schwartzman

April 13, 2018 at 4:35 AM

36 Responses

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  1. I am confident that the daisy’s mathematical competence is far greater than mine.


    April 13, 2018 at 6:16 AM

  2. This looks like flower named in my name in Latvia “Ilzīte” but it is not. Ilzīte is Anthemis tinctoria 🙂


    April 13, 2018 at 8:19 AM

  3. Thanks for the great pic of flower w/bug (leaf-footed bug?)! Also appreciate the spiral-diagrammed pix & explanations. A few days ago, we ourselves had taken a few pix of same types of flowers in our backyard.


    April 13, 2018 at 8:24 AM

  4. Beautiful flower. That looks more like summer than spring! Great photos!


    April 13, 2018 at 8:59 AM

  5. And to think Fibbonacci came about it all to explain the population growth of rabbits. It also beautifully and perfectly explains growth and balance in nature, like your daisy and seed heads and twigs on a tree. Amazing stuff.


    April 13, 2018 at 10:25 AM

    • Yes, Fibonacci came at it via highly regularized rabbits. My impression is that he wanted to exemplify the mathematical relationship and chose to embody it in rabbits for their conveniently rapid rate of reproduction. Any animal would have done; the math, that amazing math, was the important part.

      Steve Schwartzman

      April 13, 2018 at 4:27 PM

  6. If I do all this counting and data analysis, is it a nerve-ous breakdown? (Sorry!) And I cannot tell a fib, I’m grateful to Fibonacci that I don’t have to do it with Roman numerals.

    Robert Parker

    April 13, 2018 at 10:32 AM

    • That’s for sure. Actually, since I’m committed to truth-telling, the widespread assumption that the Romans calculated with Roman numerals isn’t correct. The Romans did their calculations with devices like the abacus. Roman numerals served merely to write down the starting numbers and the results. Another misconception is that the Romans normally wrote 4 as IV. They almost always wrote IIII. Ah, the things they teach us, or don’t teach us, in school.

      Steve Schwartzman

      April 13, 2018 at 4:35 PM

  7. Nice post, Steve, and I think of you whenever I inspect a sunflower! Nice tutorial as well!
    Today a friend and I will be painting another friend’s rear entrance to the house, which serves as a stairway to the upstairs apartments as well as their laundry room and back entrace to their house.. the yellow of the daisy in your image is almost the yellow that was selected. (The paint company named it ‘maracuyá’ and it definitely is the color of passionfruit juice!)

    • Inspecting sunflowers is a noble pastime. So is inspecting mathematics and its embodiments in the world.

      We could say the maracuyá juice color symbolizes your passion for painting. Now you’ll have to paint, whether on buildings or canvas or board, some exemplification of the Fibonacci numbers. You may be aware that the Spanish friars saw the parts of the passion fruit flower as exemplifying the Passion of Christ, hence the common name in English and other languages, including Peninsular Spanish:


      Steve Schwartzman

      April 13, 2018 at 4:48 PM

  8. Thanks to this pair of posts, a small mystery involving a pair of flowers may have been solved. I’m sure I found Tetraneuris scaposa on the Willow City loop. But I thought that I’d found four-nerve daisy last year, on a ranch road outside Kerrville, and those flowers were lower-growing, with leaves that continued up the stem for just a bit. Now, I’m sure that those plants — thick spreads of them that covered the roadside — were T. linearifolia. So exciting!

    None of my flowers had a cute bug visiting, though.


    April 14, 2018 at 8:18 AM

    • Another article I found this morning notes that the nerves on four-nerve daisies become darker and more prominent as the flowers age. That may help to explain the less obvious nerviness in my photos.


      April 14, 2018 at 8:35 AM

      • Let’s not forget that we intrepid nature photographers add some nerviness of our own.

        And yes, I’ve noticed that the nerves tend to darken, especially when the rays age and get papery.

        Steve Schwartzman

        April 14, 2018 at 9:10 AM

    • It’s good to have some daily excitement, don’t you think? Getting species sorted out certainly qualifies.

      Marshall Enquist says of Tetraneuris linearifolia (which he still classified as a Hymenoxys) that it’s “actually very common.” That’s been my experience.

      Steve Schwartzman

      April 14, 2018 at 9:17 AM

  9. So when I first became a cancer physician, I started doing phase I trials of new chemotherapy drugs. The concept is that you find the dose to use in later trials by escalating doses in successive cohorts of patients across a dose range. The principle is that, for conventional chemotherapy, the dose that induces toxicity relates to the dose with anti cancer effect, so you escalate until some people get toxicity. I should add that the patients have advanced refractory cancer and give fully informed consent. Anyway, each dose level is related to the previous one in a modified Fibonacci series. When I first learned of this, such was my ignorance, I went online to look up who Fibonacci is and what he’d published lately. Later when I taught on this, I used pictures of sunflower heads and pine cones to illustrate the Fibonacci series.


    April 15, 2018 at 2:56 PM

    • Thanks for giving us a great testimonial from someone who’s used the Fibonacci sequence in a practical way, not just for recreation or as pure mathematics. Thanks also for the humor in your admission that ” When I first learned of this, such was my ignorance, I went online to look up who Fibonacci is and what he’d published lately.” We share the use of pine cones as illustrations of these numbers. The house where I lived during many of the years I taught math had a pine tree on the front lawn, and I used to gather up a good supply of cones so each student had one to handle individually and count the numbers of spirals.

      Steve Schwartzman

      April 16, 2018 at 7:18 AM

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