Right from our childhood many of us may have faced difficulty in mathematics. We hated reciting tables from the primary school, solving those big equations using BODMAS rule and the geometry of the high school and not to say the calculus of pre-university. There is a popular saying in Kannada, “Mathematics is a KABBINADA KADALE” which means a groundnut of Iron! Is this perception of mathematics true? Is mathematics only filled with hard things and suffering?
‘No’ would be my answer. It is much more aesthetic than we could ever imagine. Not only it makes us think the solution to different problems in a proper way, but also many things hidden in nature. Many natural processes can be explained with the help of mathematical language. For essence, the golden ratio (1.62 approximately), is hidden in a wide variety of flora and fauna and also in many other beautiful things found in nature.
When we learn a new language, what do you think is behind the scenes? It is the Isomorphism. By proper mapping of words is how we learn different languages, keeping the known language as reference.
What is Isomorphism?
Let Rahul be a person from Delhi and Roger is a person from the USA. Rahul says “Mujhe ek kalam do” and Roger says, “Give me a(one) pen”. Both sentences are in agreement in terms of meaning but have used words of two different languages to express the same thing. These two sentences have isomorphism hidden behind them.
The term Isomorphism is derived from the Greek word Isos which means “same” or “equal” and Morphe, meaning “form”. Isomorphism can be understood clearly by mathematical tools.
In mathematics, to say any two structures/sets (The set is language in the above case) are the same, we define a bijective map from one set to another which holds certain conditions. This process of defining a map between two sets with some conditions is known as ISOMORPHISM.
By the term ‘map’ we mean a ‘rule’ that matches an element belonging to one set (defined as pre-image) to the element of another set (defined as image). This can be better understood by constructing a map between a set of students and a set of benches with ascending order of height as a rule. So each student gets a bench based on their height i.e. a map from Set 1 to Set 2.
A bijective map between two sets refers to, for any given element in one set (consider set1 of the above case), undergoing map, the output obtained is not shared with any other elements of the same set i.e. No two students share the same bench. And if we point out a particular bench from set 2, there should be a student who takes that particular bench i.e. every bench should be mapped to a student. Mathematically speaking, every element in set 2 has a pre-image from set 1 under the mapping. If these two conditions are satisfied, we say it is a bijective map between two sets. For a bijective map to exist between any two sets, they must contain the same number of elements.
With some conditions, we preserve the meaning of the words in the Rahul and Roger case. Here ‘Mujhe’ gets mapped to ‘me’, ‘ek’ is mapped to ‘one’, ‘kalam’ is mapped to ‘pen’ and ‘do’ is mapped to ‘give’. Well, that is an isomorphism. It is a mathematical process by which we can say any two structures or sets are the same by defining a bijective map between those two sets with some conditions. This is the reason why two different languages are isomorphic (similar) structures at times and not always.
The Universe is a giant brain!
A research paper submitted by an astrophysicist, Franco Vazza, and a neurosurgeon, Alberto Feletti, showed the resemblance between the two most complicated systems in nature, the brain, and the universe. When we observe these two complex systems, the brain consists of 70 billion neurons and the observable universe consists of at least 100 billion galaxies [Here, a neuron is being compared to a galaxy]. The distribution of mass is also similar, 30% of the mass of the universe is made up of galaxies, and the rest 70% with dark energy. Similarly neurons in the brain constitute about 30% of its mass and the rest 70% is of water.
The statement made by the researcher makes this topic more interesting. Franco Vazza says, “Spectral density (Technique for studying the spatial distribution of galaxies) of both the system showed that the distribution of the fluctuation within the cerebellum neuronal network on a scale of 1 micrometer to 0.1 millimeters (100 times more) follows the same progression of matter in the cosmic web but, on a larger scale that goes from 5 million to 500 million light-years”. What makes this more interesting is, it is true for any organism. And this similarity can be seen mathematically!
HOW?
As mentioned earlier, if a bijective map exists between two sets then they have the same capacity. In mathematics we say two sets are similar if there is a bijective map between them. This differs from isomorphism because to show two structures are similar, showing a bijective map between them is sufficient. It does not require special conditions as in the case of Isomorphism.
In real number system, interval is a set which contains all the numbers lying in between highest and lowest element of the set. Given any two intervals, irrespective of their range (highest number minus lowest number is range) are similar. If we consider the number of points in the interval (1, 2) i.e. all the numbers lying in between 1 and 2, and the number of points lying in the interval (1, 10) are equal.
Though they have different radius i.e. (1, 2) has radius 1 and (1, 10) has radius 9, the number of points in between them is still the same. That is their capacity is same. To understand this, if we consider a rubber string of length 1. It can be considered as interval of radius 1. And this string can be stretched to form a string of length 9 i.e. an interval of length 9.
Though the length of string has changed, the number of points on string are same as they were before stretching. Therefore, we can say that any two intervals have same capacity. Here drawing a straight line between the interval implies we are defining a bijective map. Further analyzing we can show that any interval is similar to the entire real line and the real line R is similar to the real space R^3.
So if we consider the intervals as individual brain and ℝ^3 as the cosmos, they are similar. (i.e. brain is equivalent to the entire universe). Being a tiny part of universe, brain is equivalent to entire universe. Isn’t it amazing to observe complex things with the simple mathematical language.
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