Question 1 Observe and record various mathematical patterns in nature/architecture/designs/floors and comment on their mathematical significance (Geometry).
Introduction
Shapes and Numbers have become the limelight of Mathematics-Research and now mathematics has grown from just solving sums to an experimentation and observation, finding relation to the real world and thus it involves science which brings us to conclusion, “Mathematics is the Science of Patterns, Shapes and Numbers.”
According to me, mathematics is everywhere, I can see it. In footballs, tables, the arrangement of leaves on the stem, the way a human balances their body, from tiny protons to the big universe, everything. The interesting thing is how many people just seem to ignore those facts, the facts that prove how beautiful the universe is. Shapes have always intrigued me. As a child, I used to make new shapes and name them after me. In this project, I will be portraying my love for shapes and discussing the various patterns I have noticed around me.
THE LEAVES
This is one of the most interesting patterns I have ever seen, the pattern which leaves make in order to survive and thrive, it is just beautiful.
See: The spiral pattern of an Aloe polyphylla plant at the University of California Botanical Garden.
To the untrained eye, plants may appear to grow rather impulsively, popping out leaves at random to create one big green jumble. Take a closer look, though, and you’ll find that a few curiously regular patterns pop up all over the natural world, from the balanced symmetry of bamboo shoots to the mesmerizing spirals of succulents.
In fact, these patterns are consistent enough that cold, hard math can predict organic growth fairly well. One assumption that has been central to the study of phyllotaxis, or leaf patterns, is that leaves protect their personal space. Based on the idea that already existing leaves have an inhibitory influence on new ones, giving off a signal to prevent others from growing nearby, scientists have created models that can successfully recreate many of nature’s common designs. The ever-fascinating Fibonacci Sequence, for example, shows up in everything from sunflower seed arrangements to nautilus shells to pine cones. The current consensus is that the movements of the growth hormone auxin and the proteins that transport it throughout a plant are responsible for such patterns.
Leaf arrangement with one leaf per node is called alternate phyllotaxis, whereas arrangement with two or more leaves per node is called whorled phyllotaxis. Common alternate types are distichous phyllotaxis (bamboo) and Fibonacci spiral phyllotaxis (the succulent spiral aloe), and common whorled types are decussate phyllotaxis (basil or mint) and tricussate phyllotaxis (Nerium oleander, sometimes known as dogbane).
See: An Orixa japonica shrub with the various divergence angles of the leaves visible.
Leaves on an Orixa japonica branch (upper left) and a schematic diagram of orixate phyllotaxis (right). The orixate pattern displays a peculiar four-cycle change of the angle between leaves. A scanning electron microscope image (center and bottom left) shows the winter bud of O. japonica, where leaves first begin to grow. Primordial leaves are labelled sequentially with the oldest leaf as P8 and the youngest leaf as P1. The label O marks the shoot apex.
A top-down view of leaf arrangement patterns in “orixate” phyllotaxis as new leaves (red semicircles) form from the shoot apex (central black circle) and grow outwards.
THE MATHEMATICAL BEAUTY OF SNOWFLAKES
It is so mystical when you leave in the morning in the snow. Snowflakes are flying around in the measureless universe and falling to the ground and blanketing the ground. Or a snowflake landing on your nose of its winter angel. You do not see any flowers because flowers cannot flourish in the cold. And it is so noteworthy that you come to realize that no two snowflakes are alike. It is like the uniqueness of a snowflake is so perfectly controlled by a divine force. The individuality of a snowflake’ structure is parallel to human life. Like snowflakes, everyone has a different story to tell.
Holding the snowflake in my hand and watching each angle, shape, and line melt. I am not the only one who thinks about snowflakes. A lot of mathematicians also think about snowflakes. Actually, they think about patterns because they are particularly important for three basic mathematical ideas; pattern, symmetry, and symmetry breaking.
Snow is a molecular structure of an ice crystal. And ice is a really structured substance. It is a different form of water. When the water cools down, and the molecules move more slowly, they begin to affect how the molecules line up. Hydrogen atoms of one water molecule bond with two oxygen atoms. And, as water freezes, the molecules arrange into hexagons. They prefer to stay as far away from each other as possible and that makes them take up more space. And that large space affects density. The density of ice becomes less dense than water. It means that ice floats. Almost all other liquids have higher density when they freeze.
When we examine an ice crystal profoundly under normal conditions, we always see a combination of molecules with six-fold symmetry. Snowflakes molecules make a honeycomb structure. That cause an incalculable number of hexagonal symmetries.
THE TAJ MAHAL MATH
There are not many ways someone could specifically use the Taj Mahal for math. However, the geometry used for making the Taj is used often, and frequently in all geometry classes. In the “real world” architects could use principles of the Taj to create a similar building. However, the myth is that Mughal cut off the hands of the men who built the Taj so that it would never be re-constructed. The Taj Mahal was constructed using mathematics beyond its time, as mentioned before particularly geometry. For instance, the mathematical calculations for the minarets’ weight, angles, and size, take into account the possibility of an earthquake. Not only were they built to withstand, but in the event of a serious earthquake they were constructed leaning outward, so that the fall would not affect their neighbouring building.
The astounding symmetry of the building is another aspect worth mentioning. The equal distance of windows and doors from one another, the formations of the minarets, the proportions of the domes to the arch ways. When they calculated the area and volume for the building, they only needed to measure half the actual building due to the building’s mirror symmetry. The Taj Mahal is a world-renowned monument to symmetry both inside, and out.
If an object is reflected in water many people believe that the image has a line symmetry. But it is really a “mirror image” in the case of Taj Mahal. Shah Jahan has built it with perfection to complete his love for Mumtaz.
CONCLUSION
Any of the five senses may directly observe patterns. Conversely, abstract patterns in science, mathematics, or language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature and in art. Visual patterns in nature are often chaotic never exactly repeating, and often involve fractals.
BIBLIOGRAPHY
- Medium website
- Eurekalert website
- Smithsonianmag Website
- Echos of LBI website
- Slideshare
- Docplayer
- Snehadebby weebly blog
- Google images
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