In summary, the conversation discusses the speaker's search for a mathematical function appropriate for Valentine's Day, which led them to the concept of a function and its inverse. The chosen function and its inverse create a heart shape, and the speaker notes that they have four points in common. They also provide a citation for anyone wishing to use the expressions in their work. The conversation then delves into solving a polynomial of order 4, with the speaker realizing that it is not beyond their reach when they already know two of the solutions. They then proceed to solve the polynomial, ultimately obtaining two solutions at (0;0) and (0;k-1).
  • #1
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Introduction
Being a somewhat geeky Maths ‘nerd’, I spent days leading up to Valentine’s day trying to find a Maths function appropriate to the day. In the end, I did not find an appropriate single function but (perhaps aptly enough) a pair: function and inverse function. It occurred to me that these provide a good opportunity to illustrate a number of simple mathematical transformations involving reflections in the x-axis, y-axis, and both x-axis and y-axis. Whilst the inverse function itself illustrates reflection in the line y=x.
The Function (and its Inverse)
So here is the pair of graphs I initially settled upon: $$y=-x(x-4)$$ and $$x=-y(y-4).$$ As can be seen, there is some semblance of a heart shape formed by the function and its inverse. Perhaps not quite what Valentine card designers would be looking for!
Val1.png

It intrigued me to note that this particular function and its inverse have 4 points...

Continue reading...
 

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Please suggest a citation for anyone who wish to use these expressions (e.g. for homework problems...).
 
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  • #3
Useful nucleus said:
Please suggest a citation for anyone who wish to use these expressions (e.g. for homework problems...).

Bibtex Code follows - is that ok ?

@misc{Valentine Reflections online,
author = {Neil Parker},
title = {Valentine's Reflections: Mathematical Matters of the Heart},
howpublished = {\url{https://www.physicsforums.com/insights/valentines-reflections-mathematical-matters-of-the-heart/}},
month = {March},
year = {2021},
note = {(Accessed on 03/02/2021)}
}
 
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  • #4
Thank you!
 
  • #5
"needing to solve a polynomial of order 4: y = − x ( x − 4 ) = y ( y − 4 ) ( − y ( y − 4 ) − 4 ) , or − y 4 + 8 y 3 − 20 y 2 + 15 y = 0 . Since such a task is well out of my mathematical reach, "It is not beyond your reach when you already know, as you do, two of the solutions!
 
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  • #6
epenguin said:
"needing to solve a polynomial of order 4: y = − x ( x − 4 ) = y ( y − 4 ) ( − y ( y − 4 ) − 4 ) , or − y 4 + 8 y 3 − 20 y 2 + 15 y = 0 . Since such a task is well out of my mathematical reach, "It is not beyond your reach when you already know, as you do, two of the solutions!
True enough - should have seen that one! Perhaps a little bit 'knee jerk' to send it off to WA!
 
  • #7
epenguin said:
It is not beyond your reach when you already know, as you do, two of the solutions!
Ok - better give it a go then!

$$y=-x(x-k) ; x=-y(y-k) \implies
y=y(y-k)(-y(y-k)-k)$$ $$ \implies
-y=y(y-k)(y(y-k)+k)=(y^2-ky)(y^2-ky+k)$$ $$\implies -y=y^4-ky^3+ky^2-ky^3+ky^2-k^2y$$ $$\implies 0=y^4-2ky^3+2ky^2-(k^2-1)y=y(y-(k-1))(y^2-(k+1)y+(k+1))$$ From which we obtain the two solutions at (0;0) and (0;k-1). The discriminant of the third factor quadratic is $$(k+1)^2-4(k+1)=k^2-2k-3=(k-3)(k+1)$$ in accordance with the WA solution.
 

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1. What is "Valentine’s Reflections: Mathematical Matters of the Heart"?

"Valentine’s Reflections: Mathematical Matters of the Heart" is a scientific paper that explores the relationship between mathematics and love, specifically on Valentine's Day. It discusses various mathematical concepts and their applications to love and relationships.

2. What inspired you to write this paper?

This paper was inspired by the idea of combining two seemingly unrelated topics - mathematics and love - and exploring the connections between them. Valentine's Day, being a day dedicated to love and romance, seemed like the perfect occasion to delve into this topic.

3. What are some of the mathematical concepts discussed in this paper?

Some of the mathematical concepts discussed in this paper include probability and statistics in the context of finding a compatible partner, game theory and its applications to relationships, and the mathematics behind the shape of a heart.

4. How does this paper contribute to the field of science?

This paper contributes to the field of science by showcasing the intersection of mathematics and love, and how mathematical concepts can be applied to understand and analyze relationships. It also highlights the importance of interdisciplinary research and the potential for mathematics to be applied to various fields.

5. What do you hope readers take away from this paper?

I hope readers take away a new perspective on the role of mathematics in our everyday lives, and how it can be used to understand and appreciate complex emotions such as love. I also hope this paper sparks curiosity and encourages readers to explore the connections between seemingly unrelated topics in their own research and studies.

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