This document discusses patterns found in nature that relate to the Fibonacci sequence and phyllotaxis. It provides examples of these patterns in pinecones, sunflowers, leaf and branching arrangements. Interactive models and tools are presented to illustrate these mathematical patterns. Citizen science projects are mentioned to further study these natural phenomena. The document encourages exploring these relationships across disciplines and creating STEM projects integrating art and design.

1. Seeing Math Patterns
In Nature
Mary Van Dyke, October 2020

2. Play and Patterns

3. Overview
• Pack n’ Grow
• Pinecones and Sunflowers
• Fibonacci and non-Fibonacci
• Phyllotaxis - Leaf Arrangement
and Branching
• Squares and Spirals
• Resources

4. Pack n’ Grow
Hexagonal Pack
• Maximum strength
with minimum materials
Square Pack Lattices

5. Pinecones

6. Patterns

7. How Many Spirals?

8. How Many Spirals?

9. My Pinecone

10. My Pinecone
13 spirals

11. My Pinecone
8 spirals

12. My Pinecone
13 spirals
8 spirals

13. Finding Spirals – another example
8 spirals 13 spirals5 spirals
Elements (e.g. seeds, petals, bracts, leaves) numbered in growth order

15. Who Was Fibonacci?
Liber Abacci 1202
Leonardo of Pisa – son of
Bonacci
c. 1170 – c. 1250

16. Math – The Fibonacci Series
F n = F n-1 + F n-2 where F 0 = 0 and F 1 = 1
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21

17. Pinecone Projects

18. Pineapples

19. How Many Spirals?

20. How Many Spirals?

21. How Many Spirals?

23. Many Plants Grow
in a Fibonacci Pattern

24. Grow Sunflowers

25. Sunflowers

26. Efficiency - Sunflower Spiral Patterns
1000 seeds 1000 seeds 3000 seeds

27. Elements in Growth Order
Elements (e.g. seeds, petals, bracts) numbered in growth order

28. Divergence Angle
“Golden Angle” g ≈ 137.5º
"

29. Patterns and Dynamic Models
• Wolfram Mathematica
- Interactive models
• Vi Hart
- Doodling in Math:
Spirals, Fibonacci, and Being a Plant, 3 parts on YouTube

30. Interactive Tools
Wolfram Mathematica Demonstrations
Search “phyllotaxis” for about a dozen models
http://demonstrations.wolfram.com/search.html?query=phyllotaxis
Phyllotaxis Spirals in 3D
http://demonstrations.wolfram.com/PhyllotaxisSpiralsIn3D/
On the Fundamental Theory
of Phyllotaxis
http://demonstrations.wolfram.com/OnTheFundamentalTheor
emOfPhyllotaxis
Fibonacci Tree
http://demonstrations.wolfram.com/FibonacciTree/

31. Anomalies – Why?

32. Big Data – Citizen Science
Alan Turing’s Sunflowers Global Experiment 2012

33. Sunflower Projects

34. Roast In-Shell Sunflower Seeds
Brush or pick seeds off seed-head
Soak and Salt
• Cover unshelled seeds with salt water ( ¼ to ½
cup of salt per two quarts of water
• Soak overnight
• Drain off water, pat seeds dry
(Or: Skip salting and soaking and roast unsalted)
Roast
• Preheat oven to 300 º F
• Spread seeds evenly on baking sheet
• Bake for 30 – 40 minutes
• Check and stir occasionally
• Cool before eating
• Store when cool in airtight container

35. Easy Daisies

37. Fibonacci and non-Fibonacci Series

38. Leaf Arrangement - Phyllotaxis

39. Branching

40. Branching
1
2
3
5
8
~13

41. Branching

42. Phyllotactic Architecture

43. The Pineapple, The Core and
The Gherkin…

45. Squares and Spirals

46. Squares and Spirals
1 1
2
3
5
8

47. Squares and Spirals

48. Are You Feeling Spirally Today?
https://youtu.be/14-NdQwKz9w
Vi Hart’s Doodling in Math:
Spirals, Fibonacci and Being a Plant
Part 3 of 3
6 minutes YouTube

49. The Mathematics of Leaves
https://www.u-tokyo.ac.jp/focus/en/press/z0508_00047.html
Dynamic Models of Leaves Growing

50. Take Homes
• Fibonacci-style math sequences are very
prevalent in nature and design
• Why? Efficiency for growth, leaf arrangement,
branching, flower and seed arrangement – and
related to the strength/availability of the growth
hormone auxin
• Correlations with Fibonacci-style sequences
include breeding patterns and sound/music
• You can study these patterns at all grade levels
with STEM integrated projects with art and
design components

51. Resources
http://goo.gl/HvvLdO
Credits and Resources

52. Questions?

53. Make Your Own Series
Think of a number…

54. Make Your Own Series
Think of a number…
2. Decide a second number
3. Add the two together

55. Make Your Own Series
Think of a number…
2. Decide a second number
3. Add the two together
4. Determine the next 12 numbers in your sequence
5. Name your series of numbers

56. Make Your Own Series
Think of a number…
2. Decide a second number
3. Add the two together
4. Determine the next 12 numbers in your sequence
5. Name your series of numbers
6. What are the ratios of each of your consecutive numbers
divided by the next larger number in the sequence?
7. What are your conclusions about the ratio of consecutive
numbers in your sequence – as numbers get larger?

57. Hit the Standards
Seeing Math Patterns in Nature Projects:
• Ask and Define Problems
• Plan and Carry Out Investigations
• Analyze and Interpret Data
• Develop and Use Models
• Construct Explanations and Design
Solutions
• Engage in Evidence-based Argumentation
• Use Math and Computational Thinking
• Obtain, Evaluate, and Communicate
Information

58. Thank you