Can plants teach us geometry?
Nandini D
When I looked up at the coconut tree in my garden, it appeared like a big umbrella. How big was the canopy of this tree? What could the length of one of its leaves be? The canopy is a circle! I still remember mugging up equations like C=2πr to calculate the circumference of a circle. How many geometry theorems I had to learn! It was a struggle to practice the notations and memorize them. Do plants follow some kind of measurement to grow? Will I find anything common between plant growth and geometry? I clicked pictures of a few plants and tried to measure them. My son helped me find online maths learning tools* and when I superimposed some geometrical patterns (as shown in the 12 examples) on the leaf and flower pictures I took, it was overwhelming! Why is it that we don’t teach biology and maths together? Let us see a few examples of how we can do it.
Example: 1
The ‘plane’ can be identified on any plant around us. Measuring in one direction is easy if we start with a pine needle, grass blade or an equisetum, while talking about the plant. Similarly, we can choose any broad leaf (deciduous plant) to measure 2dimension and a fallen tree trunk for 3dimension.
Example: 2
Flowers can be simple or compound. They either grow as a single flower or in clusters to attract pollinators. When we are discussing the parts of a flower and floral arrangements, can we also identify the symmetry there is? Draw an imaginary line to cut the flower into two equal halves. If you can do this only in one direction, then it is called a ‘zygomorphic’ flower. If the flower can be divided into two or more equal halves while cutting in any direction, then it is called an ‘actinomorphic’ flower. Can you make a list of simple and compound flowering plants in your garden or vicinity and check which ones are zygomorphic and which actinomorphic?
Example: 3
This dandelion clock (the seed head of a dandelion after flowering) is found everywhere in open fields. The hairy seeds, ready to fly, are arranged like a globe. All seeds arise from a single central point and have perfect symmetry in all directions. Can we gauge the radius and circumference of its circle? Are the ‘touch me not plant’ (Mimosa pudica) flowers also like this?
Example: 4
The main function of the leaf is to absorb sunlight to synthesize food. The leaves of any plant are arranged in such a way that each of them can absorb maximum sunlight. At each nodal region, the leaves are spread out such that those on top will not cast their shadows on the leaves at the bottom. The angle at which the leaves are borne on the main stem is also quite consistent at each node, to expose the leaves to sunlight. Measure the angle between the leaf stalk and the stem.
Example: 5
If you observe the blue butterfly pea plant (Clitoria ternatea) or methi (Fenugreek), you will notice that their stalks end in three leaves. How many triangles can one find at the tip of each stalk? What types of triangles do the three leaves make? Make a list of other such plants in your garden?
Example: 6
Do the leaves show symmetry on the left and right side from the point of origin? Draw a right-angle triangle on a leaf. Is it possible to teach the Pythagoras theorem?
Example: 7
Observe the leaf pattern and check whether it is a simple leaf or a compound leaf. Can you see a quadrangle on the leaf? To confirm, measure the angles on the inside.
Example: 8
Let us observe the beautiful flowers found on hedges. Ladies skirt (referred to as such because the corolla of the flower looks like a pleated ladies skirt) like ‘morning glory’ (Convolvulaceae) has five sides (pentagon) where the petals are fused (gamopetalous). In Calotropis (milk weed) the central portion of the flower is called ‘gynostegium’ which is a pentagon. The male part of flower (androecium) and female part (stigma) are fused to farm a gynostegium.
Example: 9
How many triangles and pentagons can you identify in a calotropis flower? What is the significance of milkweed in butterfly conservation?
Example: 10
Evergreen trees like conifers can survive really cold seasons. The leaves are needle like, which limit transpiration.# The reproductive structures are called cones, which are woody and cone shaped. Cones are the most significant patterns found in nature.
Example: 11
Both leaf and flower originate from a single point. By adding petals to a flower in all possible directions, it becomes many whorled and large. This creates symmetries in multiple whorls. Let us consider the growing tip of a fern which stands like a ‘?’. This is the characteristic feature of any fern plant called circinate vernation. Can we compare the Fibonacci sequence to circinate vernation? This curled leaf signifies enormous possibility for creation! The Fibonacci sequence pattern can be observed in many creations in nature@. Is it possible to compare the growth factor to a golden ratio spiral?
Example: 12
While walking in the open field, if we observe carefully, we can find many plate like outgrowths on wood. This is the ‘polypore’ fungi. Polypore is generally found on rotting wood. The fruiting body of fungi has many pores or tubes on the lower side. Does the polypore mushroom resemble a parabola? If so, can we think of measuring it?^
There are many such examples among trees, birds, butterflies. We can make a list of such observations and form connections to teach maths, chemistry, or physics.
Our eyes are only instruments to capture the picture, further persuasion of the image based on colour, shape, smell, texture matters at the individual level to understand the concept. If pedagogy can be improved by involving a sense of observation and correlating what we observe with what we teach, learning can be a joy rather than imposition. Learning maths and plants together, while walking in the garden can make for a better lesson than a mundane classroom session. Such a lesson will help children remember the terminologies and equations too. The examples mentioned here are not specific to any age group. This article suggests how different themes can be merged and taught at the same time.
*https://www.geogebra.org/calculator, https://www.mathsisfun.com/geometry/
@https://www.treehugger.com/how-golden-ratio-manifests-nature-4869736
^https://www.intmath.com/integration/6-simpsons-rule.php
The author is a botanist. She has been involved in teaching and research for more than 25 years. Currently she is working as a Consultant for the Foundation for Revitalisation of Local Health Traditions (FRLHT), Bengaluru and is a visiting faculty member at RIWATCH, Roing, Arunachal Pradesh. She can be reached at nandinidholepat@gamil.com.
