Win a Davidson Institute Fellowship: Unlock the Potential of Your Math Project

[MevsEinstein]

If you didn't know, there is something called the Davidson Institute Fellow Scholarship for middle and high school students. To get a scholarship (which come in 10,000 dollars, 25,000 dollars, and 50,000 dollars (if I remember correctly)), you will need to share a research project in the subjects provided: Technology, Engineering, Science, Philosophy, Music, Mathematics, Literature, and Outside the Box. For more information, see this link: https://www.davidsongifted.org/gifted-programs/fellows-scholarship/

Right now, I want to get the scholarship by showing one of my Mathematical discoveries. Here's what I have:

The Triangle constant ($2/(3^{1/4})$): If you multiply this number to the square root of the area of an equilateral triangle, you will get the measure of the triangle's side.

Kashi's transform: This takes in any "area of a triangle" formula and transforms it into another one that works for non-right triangles. I made a formula using Kashi's transform that contained the tangent function.

Tri-horn fractal: I messed up the generation of imaginary numbers to make this fractal: https://scratch.mit.edu/projects/596884463/ . This can mean that I have made a new way to generate fractals by ruining real math.

Gaussian Integration: This is a hypothetical integration technique derived from Gauss's "some of consecutive positive integers" formula: $n(n+1)/2$.

Since I'm supposed to have only one application, I will have to choose from one of the four. Which invention/discovery would be best to present?

 

[fresh_42]

You know that the deadline for 2022 has passed?

I haven't found any requirements of what exactly could be researched.

It is generally difficult in mathematics to find something new on a basic level. I'm therefore a bit skeptical about your fractals and integration project. Of course, you could take them and set them into the context of what is actually already known and how your ideas fit in there. But it could as well be the case, that your ideas simply do not work.

The geometric ideas are probably better, as long as you can formally prove them.

I wouldn't underestimate in general the amount of searching what is already known! Also, a view into the history of the field can be very helpful.

 

[MevsEinstein]

You know that the deadline for 2022 has passed?

Yeah I know I'm just planning to do it if I don't get accepted to CSU LA's early entrance program.

 

[MevsEinstein]

You know that the deadline for 2022 has passed?

I haven't found any requirements of what exactly could be researched.

It is generally difficult in mathematics to find something new on a basic level. I'm therefore a bit skeptical about your fractals and integration project. Of course, you could take them and set them into the context of what is actually already known and how your ideas fit in there. But it could as well be the case, that your ideas simply do not work.

The geometric ideas are probably better, as long as you can formally prove them.

I wouldn't underestimate in general the amount of searching what is already known! Also, a view into the history of the field can be very helpful.

Thanks for your suggestions!

 

[MidgetDwarf]

Yeah I know I'm just planning to do it if I don't get accepted to CSU LA's early entrance program.

Just fyi. CSULA does not normally offer Analysis 2 (Integration) and Modern Algebra 2 (Ring Theory). It is taught every two years irc. Something to consider.

 

[MevsEinstein]

CSULA does not normally offer Analysis 2 (Integration) and Modern Algebra 2 (Ring Theory).

The main reason why I wanted to join college early was to learn Integration and stuff like that. Very disappointing >:(

 

[fresh_42]

The main reason why I wanted to join college early was to learn Integration and stuff like that. Very disappointing >:(

A good start to get an overview is Wikipedia. I wouldn't learn it from there, but it is an appetizer. If you search "Calculus+pdf" or "Analysis 1 + pdf" then you will find lecture notes.

 

[pbuk]

The Triangle constant ($2/(3^{1/4})$): If you multiply this number to the square root of the area of an equilateral triangle, you will get the measure of the triangle's side.

That would be rather a short paper - it's one line of trig and 3 lines of algebra.

Kashi's transform: This takes in any "area of a triangle" formula and transforms it into another one that works for non-right triangles. I made a formula using Kashi's transform that contained the tangent function.

I don't really understand this, but if it works then can you say all there is to say about it in a couple of lines?

Gaussian Integration: This is a hypothetical integration technique derived from Gauss's "some of consecutive positive integers" formula: $n(n+1)/2$.

Note that something normally called "Gaussian quadrature" already exists as an integration technique, but $n(n+1)/2$ reminds me more of the Trapezoidal rule. I don't see how you can tackle these without calculus.

Tri-horn fractal: I messed up the generation of imaginary numbers to make this fractal: https://scratch.mit.edu/projects/596884463/ . This can mean that I have made a new way to generate fractals by ruining real math.

Phrases like "ruining real math" don't go down well in research papers, however if you are interested in fractals then this could be a suitable topic. Note however that approximately 0.01% of research papers in Mathematics involve a new "invention", that is not what research is about. Instead, research is almost always finding out about what is already known and expressing it in a new way.

Does the Davidson scholarship require that applicants are nominated by someone? I think you should discuss with a potential nominator what sort of topic would be suitable and what a successful application might look like.

 

[MidgetDwarf]

The main reason why I wanted to join college early was to learn Integration and stuff like that. Very disappointing >:(

You see integration in the calculus sequences for the first time. In analysis, you see the justification for it.

 

[MevsEinstein]

I don't really understand this, but if it works then can you say all there is to say about it in a couple of lines?

This is how Kashi's transform looks like:

$Δ(f) = f(a^2+b^2-c^2)/2ab*\cos(C) = f*2ab*\cos(C)/(a^2+b^2-c^2)$

(where $a, b$ and $c$ are sides of the triangles and $C$ is the measure of the angle opposite to side $c$)

Substituting $ab*\sin (C) /2$, for $f$, you get a new area of a triangle formula: $\tan (C) * (a^2+b^2-c^2)/2$. You can also try substituting in a random function or fraction to make a new equivalent form. For example, if you substitute $\sin (C)/c$ for $f$, you get $\tan(C) * (a^2+b^2-c^2)/2abc$.

 

[MevsEinstein]

you get a new area of a triangle formula: $\tan (C) * (a^2+b^2-c^2)/2$.

Oops I meant $\tan (C) * (a^2+b^2-c^2)/4$.