Adventures in Imperfection

Nature’s Geometry: An Introduction to Phi Pattern Paradise

“I wonder at the starry pattern in the sky
Are they little pieces of moon which want to fly..?”
Munia Khan


Calculus, algebra, and trigonometry. In high school, these were the subjects I was told (and believed) I would never master. Though my mind migrated to right-brain dalliances, I took every advanced math class offered because it was a challenge. Even when I quizzed out of college Trig and received a math scholarship, I still believed that I was “terrible at math.”

As I’ve gotten older, I can see that my self-assessment wasn’t entirely true. I am drawn to patterns, numbers, anagrams, repetition, and formulas–especially in nature.

Nature's Geometry: A Phi Pattern Paradise

Meet Mr. Phi ^^^

He’s the little dude who inspired this story. Note his (her?) shell ridges and symmetry. Now, look at the white marks scoring her (his?) body. If we were to measure the distance between each terraced ridge or patterned mark, we would find a mathematical sequence.  Does this make the impressions random or dipped in God’s calculated paint pot?

“The laws of physics is the canvas God laid down on which to paint his masterpiece”
Dan Brown, Angels & Demons

The Fibonacci Sequence, sometimes known as the Golden Ratio, is a series of numbers in which each number is the sum of the two preceding numbers.

Say again?

The simplest is the series 1, 1, 2, 3, 5, 8, etc. Want your mind blown? Look at Mr. Phi and then look at the mathematical depiction of Mr. Phi.

Mr. Phi
Mr. Phi
Mr. Phi on math
Mr. Phi on math

Random? Coincidence? If you measure your finger joints, and insert into the equation, you’ll see further proof.

{\frac {a+b}{a}}={\frac {a}{b}}\ {\stackrel {\text{def}}{=}}\ \varphi ,
2, 3, 5, 8
Art or nature?

Does knowing that nature has an algorithmic design detract from the beauty…or add to it?

Dahlia Maze
Dahlia Maze

“Science and religion were not enemies, but rather allies – two different languages telling the same story, a story of symmetry and balance… heaven and hell, night and day, hot and cold, God and Satan. Both science and religion rejoiced in God’s symmetry… the endless contest of light and dark.” ~~Dan Brown, Angels & Demons


Fractal Patterns





Each a descriptive word with a numerical depiction in nature. I’m a visual person, so if the calculator isn’t handy, I want to see it first hand.

Symmetry: For me, this is the easiest to understand. Two sides to a face, mirror images, pairs of shoes, yin and yang. It’s why we are drawn to formal flower beds, lines of buses and even integers. It’s why I notice “11:11” every time it shows up on a digital watch or phone. My brain is wired for symmetry, balance, and harmony.

A Kansas favorite, showy sunflowers boast radial symmetry as well as Fibonacci seed pods. If you look closely, you’ll notice the same sequence with leaves, petals and seeds – it’s why it’s difficult to find a four-leaf clover or poison ivy marked by anything but leaves of three.


Spirals: These arrangements have explanations at different levels – mathematics, physics, chemistry, biology – each individually correct, but all necessary together.[46] 

Fractals: infinitely self-similar, iterated mathematical constructs having fractal dimension.[17][35][36]

Thank goodness for Wikipedia–I couldn’t have come up with that definition on my own. Think of snowflakes, ocean waves, riverbeds, and craggy canyons–a method to nature’s madness.

Even in destruction and decay, fractal math is evident.

If a person tries to grasp too many patterns and formulas, thoughts often fall into sludge and chaos. This post has taken me over three months to compile, and longer to collect the photos. Even today, in my semi-organized quest to show you the beauty and review the equations for each phenomena, I find myself a bit overloaded and overwhelmed.

“Chaos was the law of nature; Order was the dream of man.” 

Henry Adams, The Education of Henry Adams

Perhaps a non-scientist/non-mathematician like me is not intended to grasp the complexities of tessellations, waves, stripes, and spheres. Maybe I’m just supposed to look and listen to the inner stirrings of a heart naturally inclined to cherish nature’s gifts. Perhaps you are, too.

And, when you look, what do you see? Do you see math, beauty…or both?


Does your head hurt a little? Worried that there was going to be a quiz?

Not now. Unless you want to guess what time this post published today…anyone?

Take a look at my Pinterest page if you want to see more beautiful patterns.

Extra treat! Mr. Phi is Instagram famous– JJ-Indetail featured him on their page this morning. He’s a shy guy, but was thrilled to see his name in lights.

Please discuss below! Mr. Phi wants to see the world through your eyes, too.

68 thoughts on “Nature’s Geometry: An Introduction to Phi Pattern Paradise

    1. FRIST!!! You da bomb, Christy. Thank you for helping me work through this – I think I just needed to walk away. I want to keep revisiting the subject in future posts – you should have seen the tesellations from the airplane this morning!!

      Liked by 1 person

  1. Mr Phi is adorable! I’d never heard of the Golden Ratio before today (btw, you’re a total mathlete in my book) and find it fascinating and loosely applicable to all things, though that’s no doubt the human craving for order. Nature is awesome and so are your pictures and words.

    Liked by 2 people

      1. It’s all natural. It’s not as mathematical as it seems. And, it’s mostly from the inside. It’s just a western view. Google wabi sabi. And, Zen thinking. The best thing any photographer can do is not think. Just like hitting a baseball. You can’t think and bat at the same time.

        Liked by 1 person

  2. i admit to getting lost, to reading a lot of “blahblahblah”….

    but, to me, the pictures tell the story and if i don;t read the words i get it, does that make sense?
    I am a visual learner though…and these were excellent examples!

    Liked by 1 person

  3. My head was hurting until you said there wasn’t going to be a quiz! I always believed I was horrible at math but I know now that it was just because I had so much stacked up against me in high school and I couldn’t concentrate. I see math and beauty when I look at nature. Right now I’m watching rain gently drop into my swimming pool and it’s too perfect to be random but how can raindrops be planned? Such a fun and interesting post my friend! ❤

    Liked by 1 person

    1. I LOVE your raindrop application. I was trying to capture rain drops on the airplane window today because they were so chaotically…organized! Christy promised me that her head did not hurt…even though mine totally did after writing this. Thank you for reading and indulging me – I think it’s time for something light next time around 🙂

      Liked by 1 person

  4. MM this is GORGEOUS and I love your carefully compiled photographs and examples. I really *am* terrible at maths but I adore the Fibonacci sequence (I have the numbers tattooed between my shoulder blades, in the spiral) and LOVE seeing nature and maths working so beautifully together. There’s something about the precision which is utterly, indescribably beautiful.

    One of the most profound moments of my life was seeing the engine start from still, on the Waverley, the world’s last ocean-going paddle steamer. She had these ENORMOUS brass pistons and her engine filled a huge amount of space, all fiddly and intricate and it gradually began moving and speeding up until the paddles were whipping the sea into a foam inside the paddle-boxes, and the parts of the engine were flinging around, brass gleaming gold sparkles and lustre everywhere, whizzing at a million miles an hour in ABSOLUTE precision…it was incredible.

    I loved it.

    P.S. Snails are hermaphrodites.

    Liked by 1 person

    1. Dearest Lizzi…I am blown away! How did I not know about your tattoo?? And your description abou the Waverly! Sublime! I am a closet steampunk fan…your description made me feel like I was there. And I had forgotten that snails were hermaphrodites. I am TOTALLY consulting you for my next science post 🙂

      Liked by 1 person

      1. *GRINS* I can now admit I was a *little* surprised you didn’t know…it seemed to me like so much the kind of thing you WOULD know! I’m glad you did 🙂

        If you’re a steampunk fan or fan of ANY kind of machinery then I strongly urge you to see if you can get a ride on a paddle steamer with an open engine. It’s utterly, UTTERLY beautiful. There are videos on Youtube of Waverley’s, but it’s just not the same as being there in person.

        And…I have NO idea how you didn’t know about the tat, but that kind of brings me glee. I know it’s in a place I never see it bar in the mirror, but I LOVE having it and knowing it’s there 🙂

        Liked by 1 person

      2. I think maybe I’m senile! Perhaps I knew about your tattoo and then forgot? Oh me, it seems that I forget more than I remember these days. Thank you for being here, sweet Lizzi. Every time I think I’m coming back and re engaging, life gets in the way again. I hope you’re doing well…I can’t wait for you to be here. xoxoxo

        Liked by 1 person

      3. Life does that, my lovely. It’s okay. Trust to timing, and I guess to the knowledge that your writing is so a part of you it will never fade entirely.

        I’m doing okay, surprisingly okay at the moment, which is a huge relief, in spite of everything. I am sending wishes for a more relaxed time for you, though, dear MM ❤ ❤

        And yeah – SO can't wait til I'm there!

        Liked by 1 person

  5. Yep my brain hurts now… thanks bunches! LOL! *groan*
    But, having said that, I totally understand being drawn to patterns. I need structure and routine in my life and a pattern helps me to learn and retain information. It is easier for me to learn lyrics to a song than it is to learn lines in a play. There is a pattern. There is also critical thinking in mathematics and I approach life somewhat with critical thought. So I find logic in math, I just don’t like to do it.
    This was a beautiful post! Thank you so much for taking the time to put it together to help the rest of us see a small glimpse into that fabulous brain of yours and the things you find perfect in nature!
    How can people say there is no God???? 🙂 ❤


  6. I do enjoy seeing math in nature because I never thought any of it would be relevant to my life. I am leery of Dan Brown though. I think he believes some of the stuff he proposed in The DaVinci Code. He scares me a little.

    Liked by 1 person

  7. I know a certain brit who has the Fibonacci spiral tattooed on her back! Love love love that math is reflected in science, nature, the cosmos. etc etc. It’s fascinating. what gorgeous photos! This post is a piece of art. xoxoxo


  8. Published at 1.13pm….because you forgot a bit and had to go back and fix it 😀 , and missed the 1.11pm numbers 😀
    But regardless, you are a jewel Michelle…no wonder you love nature 🙂
    Great post, it is good to ‘see’ you from another angle (more numbers 😀 ).


      1. Actually Michelle, for some bizarre reason I have written the time as 1.11pm (1.13pm jokingly), instead of what you had even written was your ‘numbers’ 11.11 😀
        That is strange….or maybe its just the time difference between here and there, another one of those calculations that has so much meaning within 😀

        Liked by 1 person

  9. Being a visual person isn’t as easy as it sounds. I envy those peeps to whom math comes without need of a couple vicodin. Math does open up an entirely new world of possibilities, indeed. My calculations are visceral from the get, after which I do the math on whether I want to add or subtract from my personal ledger. I get down with the math of things the same way I do most tasks I’m not crazy about- with much bitching and a vast arsenal of four letter words.

    The pictures talk to me, and these spoke kindly and beautifully. Unlike the voices in my head who speak loudly and make me want to file for a restraining order- if such a thing was even possible. But I digress….

    The pictures are kind and beautiful.

    Liked by 1 person

    1. If you find a way to get a restraining order on those voices, please let me know! Mine are extra loud tonight. I have fiddled and fiddled with this post and still wasn’t happy with the end result. Christy was the person who gave me the final push and said all was good. I was honestly overwhelmed by numbers and formulas and inadequacy – like high school all over again.

      All that aside. DUDE! I loved seeing you here today. Thank you for taking the time to read and comment. You know it means the world to me. Xo

      Liked by 1 person

  10. Gregory Bateson observed that only living things create static spirals (like snail shells and flowers) which record the story of their growth, and all the spirals found in the inorganic realm are matter in motion, from galaxies to water going down a drain. Lovely post.


    1. Thank you, sweet Vinny! I would have done more (on the tessellations and meanders) but I’m still collecting photographs. I hope you’re having a great week!
      I totally need to pop over and see what you’ve been up to!

      Liked by 1 person

      1. You did a lot.. Thats a lot of research and exposition, you just did. I couldnt do that. Too much

        Thank you. Im having an excellent week. I hope youre having a great one yourself.

        Oh.. you can take your time if you wish.

        Liked by 1 person

  11. Yeah – my head hurts. It doesn’t seem to matter how many times I read about the Fibonacci Sequence, it slides out of my head like butter on hot Teflon. The finger joint was pretty cool though.
    Nature is truly a glorious thing.

    Liked by 1 person

  12. Well,well,well Mama, chicken soup for the soul. Delicious. perfect. I have to be honest, I did not expect this from my beautiful, artistic, people blogger. I too love this math and am certain it leads to God. Well Done Mama.

    Liked by 2 people

  13. I see the beauty and sense the math. I know that the math is there. But, I look at nature and think of magic — I know there’s a “trick” behind it, something solid and unwavering, but beauty and breath-taking illusion makes me feel like anything is possible. Lately, I’ve been obsessed with sunflowers.

    I flunked algebra in 9th grade and had to take it in summer school. That was the summer I learned to play pool instead, flunking algebra for a second time. Unfortunately, my dad didn’t buy my reasoning that playing billiards was all about using math, therefore my summer school experience should be measured as a success.

    Liked by 1 person

    1. Dear Mary,
      Your comment is like poetry – I love that feeling of “anything is possible.”
      I also adored your 9th grade algebra story. I totally think that billiards counts. TOTALLY!


    1. My mind hurt when I finished. In fact, I may have stopped short because I just couldn’t do all the maths anymore.
      I hope you’re doing well sweet friend – I see that you have a post waiting in my inbox. xoxox
      Love from Kansas


  14. 59th! If only we could all see math as a beautiful flow or shell or bubble. We could find it in the most unlikeliest of places, like a Mona Lisa painting. I completely respect the amount of research put into this post. The photos are equally amazing.

    Liked by 1 person

    1. Hi Eric! How awesome to see you here 🙂 I totally forgot that you are a math guy (Pi day!!). Admittedly, the post was starting to get away from me. I wanted to spend more time on meanders and tessellations, but my brain was fried. Maybe in the future…
      October is my last busy month, and then I hope to get caught up with all my friends. How are Crash and Burn?

      Liked by 1 person

      1. It’s tough to write with a fried brain 🙂 I’ll keep an eye out for meanders and tessellations. Crash and Bang are doing great. Bang is becoming a prolific reader and Crash still loves his Minecraft!

        Liked by 1 person

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