Mathematics at Home

Dear Madams and Sirs. Hope all is well. So far, I have been reading articles and newsletters from various people. Thank all for sharing valuable thoughts. I thought it is time I wrote an article and share with all, and I welcome all to read and share thoughts, comments and suggestions to improve writing. Happy reading!

The article is titled "Mathematics at Home", dealing with observations of certain common processes, phenomena and equipment in home and deriving mathematics from them. In the book "Mathematical Principles of Natural Philosophy", Sir Isaac Newton explains how natural phenomena can be translated to mathematical equations. With an intention to understand the origin of equations and driven by curiosity to understand the patterns followed by motions of nature, I thought I could observe and study the patterns and translate them as well.

I have made a series of observations and found that principles like graphs, probability, permutations and combinations, differentiation and integration and matrices are hidden at home! There are discussed one by one.

Observation of a Glowing Stick

Once my parents placed a glowing stick on a table in the room where I work to keep the room atmosphere pleasant. This practice continued everyday. I observed the glowing stick. The flame gradually moved down with the ash above. The ash began falling down once the flame reached a certain distance, with repetition of the process until the entire stick is consumed and with ash left. I was curious whether the ash fall follows a random or definite pattern. To my surprise, the fall indeed followed a definite pattern. The fall happened at regular intervals of time, with 12 minutes per fall when observed for the first time and 9 minutes per fall when the stick was in an inclined position. One day, I switched on the fan to see the interval of fall under turbulence. As expected the falls happened at quicker intervals of time, with 5 minutes per fall. The initial time was 6:16-6:17 AM. The time of first fall was 6:21 AM; second, 6:26; third, 6:31; fourth, 6:36; and till the end, 6:38 AM. So the falls are at exactly 5 minutes interval. Pictorically, the process is shown below.

Next time, I changed the regulator speed and observed the glowing stick to see the interval of fall under reduced turbulence. As expected, the time interval is intermediate between room atmosphere and fan at full speed, with a 7-minute interval. With t1, t2, t3 and t4 as the time intervals and p1, p2, p3 and p4 as the speeds of fan (regulator at full speed and positions 2, 3 and 4, respectively), there are different permutations and combinations, which are shown below.

Differentiation and Integration from Banana and Lady's Finger

When we look at a banana after having a bite or a cut piece of banana in a fruit salad, we observe that there are six seeds visible, with two seeds at the vertices of an equilateral triangle. If we take the length of the cut piece as the differential length, dl, and extrapolate along the length of the banana, we get the total number of seeds of the banana, as shown below.

The same principle can be applied for similar fruits or vegetables like lady's finger.

Understanding e^x, e power x, or e to the power of x from Tail-End Rotation of a Fan

In schools and colleges, we come across exponential curves in text books. We have learnt the value of e lies between 2 and 3. But how are the exponential curves derived? Do such patterns exist in nature? Why is the value between 2 and 3 and not between 3 and 4 or between 1 and 2, which virtually refers to the same thing? We understand that there is a pattern where 1 becomes 2, 2 becomes 4, 4 becomes 8 and so on, and this pattern is called 2^x, similar to an enzyme or microbe division. But we need to understand e^x.

Let us consider the rotation of a fan. When the fan is stopped, it is obvious that the fan does not stop immediately but stops after a while, what is called as tail-end rotation. I tried to understand the pattern of the tail-end rotation. From the time when the rotations become visible to the naked eye, I started determining the rotations per second (rps) of the fan. The time taken for the first 5 rotations is 4 seconds; for the next 5 rotations, the time is 5 seconds; for the third, 6 seconds, fourth, 7 seconds; fifth, 9 seconds; sixth, 13 seconds; and for the last two rotations, the time taken was 26 seconds! The pattern thus becomes exponential. Ours is a three-bladed Usha Fan, the dimensions and the pattern of tail-end rotation of which are shown below.

Matrices from Triangular Chips

My parents used to offer spicy food items like chips while having lunch or dinner sometimes. One of the varieties served is the triangular chips, as shown below.

While eating, I used to count the holes of the chips, both front and rear. The holes are arranged in rows, as shown above, with the first row having 5 holes, second, 7 holes, and so on. Similarly, the back side has holes arranged in rows. If we put this in matrix form, we get a matrix shown below, with a particular case taken out of the number of cases observed or collected.

The matrix is | 5 9 10 9 6 5 3 2 |

| 4 7 10 9 7 6 4 2 |

which is a 2x8 matrix (the matrix brackets appear broken; normal brackets are used since I do not know how to draw a matrix here, sorry for this)

Naming the matrix as A, if we take the transpose, AT, and multiply by A, we get an interesting pattern, as shown below.

Summation and Integration of Knit Flowers

Similar to the cases of banana and lady's finger, we have the case of knit flowers, this time the principle of summation involved. Below is the picture of the flowers (called 'Malli Poo' in Tamil). There is a practice of knitting two flowers in a knot usually, with the distance between two knots approximately 1 cm or the width of a finger. In the practice, sometimes, flowers slip out of fingers, and so, a knot may have one instead of two flowers. Overall the flowers in the knots may be 1 or 2, with higher numbers not possible. Below are the calculations.

Conclusion

So there are a number of natural phenomena that we can observe in our day-to-day life that can be translated into mathematical equations.

"Nature to him (Sir Isaac Newton) is an open book, whose letters he could read effortlessly", remarked Sir Albert Einstein on Sir Isaac Newton. How true!