Waste less time on Facebook — follow Brilliant.
×

An interesting Number

I found this number quite a while ago just messing around with simple algebra and trigonometry. I am pretty sure it is too simple to have not been found before me, but it has some interesting qualities to it.

It is found by constructing the golden rectangle and drawing a line segment to BF and finding its length (this is not the arc illustrated in the image). If the number is 'b': \[b = (\sqrt{(5-\sqrt{5})/2}) \approx 1.17557\] There are some relationships between it and the golden ratio of course: \[b = \frac{\sqrt{(\varphi x-x)^{2}+x^{2}}}{x}\]

\[\Phi = \sqrt{(b^{2}-1)}\]

\[4(b^{2}-1)=(\frac{2}{\varphi})^{2}\] And there is one last relationship which is, at least to me, very interesting. We can represent the golden ratio as \(2cos(36^{\circ})\). And what is interesting is that, using simple math, you can get this relationship:

\(2\cos(36^\circ) = \frac{\sqrt{5}+1}{2}\)

\(\cos(36^\circ) = \frac{\sqrt{5}+1}{4}\)

\(\sin(36^\circ) = \frac{x}{4}\)

\(4\sin(36^\circ) = 2b\)

Finally:

\(\boxed{\varphi^{2}+b^{2} = 4}\)

Which leads to an interesting question, which I cannot prove, but seems to be the case. It seems that 'b' is the only constant which gives a rational answer to the equation: \[\theta = \tan^{-1}\left ( \frac{\varphi}{\sqrt{n^{2}-\varphi^{2}}} \right )\] If theta is measured in degrees and \(n \geq 2\). It is also the smallest whole solution to this equation.

Anyway, that is the strange number; I hope it is interesting to someone.

Note by Drex Beckman
1 year, 8 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

I find this very interesting but I am unable to understand what is \(x\).

Akshay Yadav - 1 year, 8 months ago

Log in to reply

Just the length of the square. You can just set them as one. For any real x, you should get the same number.

Drex Beckman - 1 year, 8 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...