WEEK 1
(Mathematics in our World)
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UNIT I. MATHEMATICS IN OUR WORLD by Gian Carlo S. Gaetos
Overview
Mathematics is the study of the relationships among numbers, quantities, and shapes. It includes arithmetic,
algebra, trigonometry, geometry, statistics, and calculus. Mathematics nurtures human characteristics like the power of
creativity, reasoning, critical thinking, and others. It provides the opportunity to solve simple and complex problems in many
real-world contexts using various strategies. Mathematics is a universal way to make sense of the world and communicate
the understanding of concepts and rules using mathematical symbols, signs, proofs, language, and conventions.
Mathematics helps organize patterns and regularities in the world. The geometry of most patterns in nature can be
associated directly or indirectly to mathematical numbers. Mathematics helps predict the behavior of nature and phenomena
in the world. It helps control nature and occurrences in the world for the good of mankind. Because of its numerous
applications, mathematics becomes indispensable. Mathematics, being a science of patterns, helps students utilize,
recognize, and generalize patterns that exist in numbers, shapes, and the world around them. Students with such skills are
better problem solvers and have a better sense and appreciation of nature and the world. Hence, they should have
opportunities to analyze, synthesize, and create a variety of patterns and to use pattern-based thinking to understand and
represent mathematical and other real-world phenomena. These explorations present unlimited opportunities for problem-
solving, verifying generalizations, and building mathematical and scientific competence.
Learning Objectives:
At the end of this lesson, you are expected to:
1. Identify patters in nature and regularities in the world;
2. Articulate the importance of mathematics in one’s life;
3. Argue about the nature of mathematics, what it is, how it is expressed, represented and used; and
4. Express appreciation for mathematics as a human endeavour.
Setting Up (Unit 1)
jefferson Lamosa
Name: _______________________________________________________Score: ___________________________/25
ACT-1B
Course/Year/Section: ___________________________________________ Date: _______________________________
Look around. Do you see trees? How about other plants and animals? What is the weather today? Do you see clouds?
Draw/capture and paste specific field of view outside of your room or home and tell or name all the pattern/s that you can
see. Tell something about the pattern you observe. Present at least three but not more than five observations.
Lesson Proper
Patterns in Nature
Pattern that you can see in nature are visible regularities or form found in the natural world. These patters persist in different
contexts and can sometimes be demonstrated mathematically. These include symmetries, trees, spirals, meanders, waves,
foams, tessellations, cracks and stripes1.
Living things like plants and animals such as orchids, hummingbirds and peacock’s tail have abstract designs with a beauty
of form, pattern and color that artists struggle to match 2. The beauty that people perceive in nature has causes at different
levels notably in he mathematics that governs what patters can physically form, and among living things in the effects of
natural selection, that govern how patterns evolve.
Mathematics seeks to discover and explain abstract patterns or regularities of all kinds 3. Visual patterns in nature find
explanations in chaos theory, fractals, logarithmic spirals, topology and other mathematical patterns. In Biology, natural
selection can cause the development of patterns in living things for several reasons, including camouflage 4, sexual selection
and different signaling, including mimicry and cleaning symbiosis. In plants, the shapes, colors and patterns of insect-
pollinated flowers like the gumamela (Hibiscus rosasinensis Linn.) have evolved to attract insects such as bees. Radial
patterns of colors and stripes, some visible only in ultraviolet rays serve as nectar guides that can be seen at a distance 5.
Types of Pattern
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Symmetry
Symmetry is general in all living organisms. Animals mainly have bilateral or mirror symmetry, as do the leaves of
the plants and some flowers such as orchids6. Plants often have radial or rotational symmetry, as do many flowers
and some groups of animals such as sea anemones. Fivefold- symmetry is found in echinoderms, the group that
includes starfish, sea urchins and sea lilies 7.
Among non-living things, snowflakes have striking six-fold symmetry. Each flake’s structure forms a record of the
varying conditions during its crystallization, with nearly the same patter of growth on each of its six arms 8.
Figure 1. Five-fold symmetry of an echinoderm Source:
https://en.wikipedia.org/wiki/Starfish
Figure 2. Radial or rotational symmetry of a cactus
Source: https://www.pinterest.ph/pin/29414203785745129/
Figure 3. Six-fold symmetry of a snowflake
Source: Mariia Tagirova/Shutterstock
Trees, fractals
Fractals are infinitely self-similar, iterated mathematical constructs having fractal dimension 9. For
example, the leaves of ferns and umbellifers (Apiaceae) are only self-similar (pinnate) to 2, 3 or 4 leaves.
Fern-like growth patterns occur in plants and in animals including bryozoan, corals, hydrozoa, and others.
Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds, river networks, fault lines,
mountains, coastlines, animal coloration, snow flakes, crystals, blood vessel branching, actin cytoskeleton,
and ocean waves10.
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Figure 4. Romanesco Broccoli exemplifying fractal pattern
Source: https://www.growseed.co.uk/cauliflower-seeds/romanesco-cauliflower.html
Figure 5. Clover leaf
Source: https://www.amazon.com/slp/clover-leaf-plant/s2683d3kprrr3u9
Spirals
Spiral symmetry is also largely present in nature. To name a few, we find spirals in sunflowers,
nautilus shells, snail shells, and weather patterns. A spiral is formed because of a property of growth known
as self-similarity, which means that the same shape is maintained as the creature grows.
Figure 6. Nautilus shell
Source: https://www.netclipart.com/isee/ibmRbbx_nautilus-shell-transparent-background/
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Figure 7. Sunflower
Source: https://www.treehugger.com/nature-blows-my-mind-hypnotic-patterns-sunflowers-4859272
Chaos, meanders, flow
Chaos, meanders and flow have simple patterns in nature with a chaotic behavior, whirling patterns
and the like. Meanders are sinuous bends in rivers or other channels, which form a fluid, most often water,
flows around bends. As soon as the path is slightly curved, the size of the curvature of each loop increases
as helical flow drags material like sand and gravel across the river to the inside of the bend.
Figure 8. Textile Cone
Source: https://en.wikipedia.org/wiki/File:Textile_cone_(cropped).JPG
Figure 9. Brain coral (Diplora strigosa)
Source: https://en.wikipedia.org/wiki/File:Diplora_strigosa_(Symmetrical_Brain_Coral)_closeup.jpg
Wave, dunes
Waves are disturbances that carry energy as they move. As waves in water or wind pass over
sand, they create patters of ripples. When winds blow over large bodies of sand, they create dunes.
Figure 10. Boelge stor (Waves)
Source: https://en.wikipedia.org/wiki/File:Boelge_stor.jpg
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Figure 11. Afghanistan (Sistan) Wind Riffles (Dunes)
Source: https://en.wikipedia.org/wiki/File:1969_Afghanistan_(Sistan)_wind_ripples.tiff
Bubbles, foam
A soap bubble forms a sphere, a surface with minimal area – the smallest possible surface area
for the volume enclosed. Foam is a mass of bubbles; foams of different materials occur in nature. At the
scale of living cells, foam patterns are common; radiolarians, sponge spicules, silicoflagellate exoskeletons
and calcite skeleton of a sea urchin.
Figure 12. Bubbles
Source: https://en.wikipedia.org/wiki/File:Equal_spheres_in_a_plane.tif
Figure 13. Foam
Source: https://en.wikipedia.org/wiki/File:Foam_-_big.jpg
Tessellations
Tessellations (repeating tile patterns or tilling) is a pattern made up of one or more geometric
shapes joined together without overlapping or forming any gaps to cover a plane. The spots on giraffe’s
body also exhibit a tessellation. Naturally occurring tessellations can also be found in snake’s skin, turtle’s
shell, lizard’s scales, dragonfly’s wings, spots of a tiger, and so on.
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Figure 14. Honeycomb
Source: https://en.wikipedia.org/wiki/File:Kin_selection,_Honey_bees.jpg
Figure 15. Fritillaria meleagris blomst
Source: https://en.wikipedia.org/wiki/File:Fritillaria-meleagris-blomst.JPG
Cracks
Cracks are linear openings that form in materials to relieve stress. When an elastic material
stretches or shrinks uniformly, it eventually reaches its breaking strength and then fails suddenly in all
directions, creating cracks with 120 degrees joints, so three cracks meet at a node. The pattern of cracks
indicates whether the material is elastic or not.
Figure 16. Cracked earth in the Rann of Kutch
Source: https://en.wikipedia.org/wiki/File:Cracked_earth_in_the_Rann_of_Kutch.jpg
Figure 17. Palm tree bark pattern
Source: https://en.wikipedia.org/wiki/File:Palm_tree_bark_pattern.jpg
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Spots, stripes
Leopards and ladybirds are spotted; anglefish and zebras are striped. These patterns have an
evolutionary explanation: they have functions, which increase the chance that the offspring of the patterned
animal will survive to reproduce. One function of animal pattern is camouflaging, for example, a leopard
that us harder to see catches more prey. Another function is signaling – for instance, a ladybird is less likely
to be attacked by predatory birds that hunt by sight, if it has bold warning colors, and is also distastefully
bitter or poisonous, or mimics other distasteful insects.
Figure 18. Leopard
Source: https://en.wikipedia.org/wiki/File:Leopard_africa.jpg
Figure 19. Cropped Photograph of Array of Ladybirds by G. G. Jacobson Source:
https://en.wikipedia.org/wiki/File:Jacobs24.jpg
Numbers in Nature
We have seen that patterns are everywhere in nature. But do you know that some of these patterns can be explained
mathematically? Yes, there is a link between mathematics and some patterns found in nature. The Fibonacci sequence, for
example, appears very often in flowers, such as in the number of petals, the arrangement of seeds, or in the arrangement
of leaves around a stem.
Below are sunflower seed pattern. You can easily spot two families of spirals, one running counter-clockwise and
one running clockwise direction. The number of spirals is 34 and 55, respectively. Some sunflowers have 21 and 24 spirals;
some have 55 and 89 or 89 and 144 depending on the species. Notice that such a pair of number of spirals actually forms
two consecutive numbers of the Fibonacci sequence.
Figure20. Sunflower Head Pattern with Counter-clockwise Spirals
Source: https://momath.org/home/fibonacci-numbers-of-sunflower-seed-spirals/
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Figure 21. Sunflower Head Pattern with Clockwise Spirals
Source: https://momath.org/home/fibonacci-numbers-of-sunflower-seed-spirals/
Another example in which we can find Fibonacci numbers in nature is in the number of petals different flowers have. Some
lilies and iris have three petals, gumamela and calachuchi have five, some variety of sampaguita has eight, corn marigolds
have 11, and some daisies have 34, 55 or even 89 petals. Fibonacci numbers also appear in the arrangement of leaves
and branches in some plants.
Nature is full of shapes and patterns. Look closely to things around you such as flowers, trees, animals, rocks, rivers, etc.
Try to search for patterns so that you may be able to find mathematical correlations to this.
The Golden Ratio and the Fibonacci Numbers
Leonardo Pisano Bogollo lived between 1170 and 1250 in Italy. His nickname, “Fibonacci” roughly means “Son of
Bonacci” (Fibonacci Sequence, 2016).
Aside from being famous for the Fibonacci Sequence, he also helped spread Hindu Arabic Numerals (0, 1, 2, 3, 4, 5, 6, 7,
8 and 9) through Europe in place of Roman numerals (I, II, III, IV, V, etc.). Fibonacci Day is November 23, as it has the digits
“1, 1, 2, 3” which is part of the sequence, which he developed. This famous Fibonacci sequence has fascinated
mathematicians, artists, designers, and scientists for centuries. Also recognized as the Golden Ratio, the Fibonacci
sequence goes like this:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 …
What is the value of the next Fibonacci Number?
Example: Fib (n) = x ; Fib (2) = the 2nd Fibonacci Number, thus Fib (2) = 1.
Another example: Fib (7) = 8 ; Fib (12) = 89 ; Fib (1) = 0
Each number in the sequence is the sum of the two numbers, which precede it. The ratio of any two successive
Fibonacci numbers is very close to the Golden Ratio, referred to and represented as phi (ϕ) which is approximately equal to
1.618034… . The bigger the pair of Fibonacci numbers considered, the closer is the approximation.
A B B/A = ϕ
2 3 1.5
3 5 1.66666666666667
5 8 1.6
8 13 1.625
…
13 …
21 …
1.615384615384615
144 233 1.6180555556
233 377 1.6180257511
…
377 …
610 …
1.618037135278515
75025 121393 1.6180339887
121393 196418 1.6180339888
196418 317811 1.6180339887
…
317,811 …
514,229 …
1.618033988754323
Try to solve the missing patterns in columns A, B and B/A below:
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Figure 22. Fibonacci Rectangle and the Golden Ratio
Source: https://noughtyscience.wordpress.com/2015/07/20/fibonacci-from-a-simple-sequence-to-the-golden-ratio/
In geometry, a golden spiral is a logarithmic spiral whose growth factor is phi, the golden ratio. That is, a golden
spiral gets wider (or further from its origin) by a factor of phi for every quarter turn in makes. Examples of Fibonacci flowers
are: three-petal lily and iris; five-petal wild rose, larkspur, buttercup and columbine; eight-petal delphiniums; thirteen-petal
flower ragwort, corn marigold and cineraria; 21-petal aster, chicory and black-eyed susan; 34-petal phtethrum and plantain
and others
References
The following are the references used to discuss several pieces of information in the discussion. References are based on
the order of appearance or citation, not in alphabetical order. Sources for figures used in this module are cited below the
figures.
1. Stevens 1974, p.3 cited at https://en.wikipedia.org/wiki/Patterns_in_nature#CITEREFStewart2001
2. Balaguer, Mark (7 April 2009) [2004]. "Platonism in Metaphysics". Stanford Encyclopedia of Philosophy. Retrieved
4 May 2012.
3. Steen, L.A. (1988). "The Science of Patterns". Science. 240: 611–616. doi:10.1126/science.240.4852.611
4. Darwin, Charles. On the Origin of Species. 1859, chapter 4.
5. Koning, Ross (1994). "Plant Physiology Information Website". Pollination Adaptations. Retrieved May 2, 2012.
6. Stewart 2001, pp. 48–49. Cited at https://en.wikipedia.org/wiki/Patterns_in_nature#CITEREFStewart2001
7. Stewart 2001, pp. 64–65. Cited at https://en.wikipedia.org/wiki/Patterns_in_nature#CITEREFStewart2001 8.
Stewart 2001, p. 52. Cited at https://en.wikipedia.org/wiki/Patterns_in_nature#CITEREFStewart2001
9. Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan.
10. Addison, Paul S. (1997). Fractals and chaos: an illustrated course. CRC Press. pp. 44–46.
Guide Question for you to ponder:
What can you say that everything you see in nature is governed by the patterns based on mathematical concepts and
theories? What is it about mathematics that might have changed your thoughts about it? Write your thoughts on the sheet
provided after the unit assessment.
CHOOSE ONLY ONE ACTIVITY between Assessing Learning Unit 1-A and Assessing Learning Unit 1-B. You don’t need
to answer both of the activities.
Assessing Learning (Unit 1-A)
Show your appreciation in the short movie and article inspired on numbers, geometry and nature by writing an essay to
answer the question: What have you learned from this video/article?
Short movie: Nature by Numbers by Cristobal Vila https://vimeo.com/9953368
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Article: Spiral and Golden Ratio by Gary Meisner https://www.goldennumber.net/spiral
Assessing Learning (Unit 1-B)
Prepare a 2-page synthesis paper focusing on ONLY ONE of the following aspects of mathematics. You may use
several articles but please cite responsibly and correctly using APA style of citation.
1. Mathematics helps organize patterns and regularities in the world.
2. Mathematics helps predict the behavior of nature and phenomena in the world.
3. Mathematics helps control nature and occurrences in the world for the good of mankind.
4. Mathematics has numerous applications in the world, making it indispensible.
Assessing Learning (Unit 1)
Name: _______________________________________________________Score: ______________________________
Course/Year/Section: ___________________________________________ Date: _______________________________
Answer to Assessing Learning Unit 1- _____
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Answer to the Guide Question: What can you say that everything you see in nature is governed by the patterns based on
mathematical concepts and theories? What is it about mathematics that might have changed your thoughts about it?
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Assessing Learning Unit 1 References:
