a^2 + b^2 = c^2

#sailing #boat #boats #math #pythagorean #photography #water #Pentax #PentaxK70

Been thinking quite a bit about the connection between numerology and astrology lately...

Tried to compile a table to help me get a sense of what's what.

But man, there are sooo many variations on the number assignment! 🤦‍♀️

No doubt because of the many different numerology systems (chaldean, pythagorean, hebrew etc) and the introduction of modern planets into the mix. 🪐

Be interested to hear people's opinions about what numbers they find work best for each planet, especially the ones that tend to move around a lot like Mars and Saturn. :mars_photo:

I am toying with the idea of the earth being the actual planetary ruler of Taurus - makes a lot of sense to me, given the strong connection to the four elements. 🌏

Also have to say that I was surprised to find Mercury associated with the number 5...but then I guess it is the odd one out with regards to its gender fluidity....

#astrology #numerology #chaldean #pythagorean
#herbrewalphabet #kabbalah

One day, one decomposition
A009177: #Numbers that are the hypotenuses of more than one Pythagorean triangle

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A009177.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A009177.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #hypotenuses #Pythagorean #triangle #graph

One day, one decomposition
A009112: Areas of Pythagorean triangles: numbers which can be the area of a right-angled triangle with integer sides

3D graph #threejs #webGL ➡️ https://decompwlj.com/3Dgraph/A009112.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A009112.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #area #Pythagorean #triangles #graph

Euclid's proof of the Pythagorean Theorem is shown in the attached figure.

You can learn more about the Pythagorean Theorem in my recent writing: https://medium.com/illumination/learning-the-pythagorean-theorem-with-an-intuitive-approach-ecd794529cf5

#math #Pythagorean #Medium #Euclid

Another fascinating Mathologer video, about profound connections between #Fibonacci numbers and #Pythagorean triangles

https://youtu.be/94mV7Fmbx88

My #mathematics bar trick is that I can prove the #Pythagorean theorem in 30 seconds.

That for a right triangle with hypothenuse c it holds that a² + b² = c², where a and b are the two other sides.

I wonder if the Babylonians, who probably knew the theorem a thousand years before Pythagoras, also used the identity to show off after a drink or two.

What's your favourite bar trick?

Prof. M. David Litwa segment on Saturninus of Antioch, his series on #Gnostic, #Platonist, #Pythagorean or #ChristianMystics of antiquity. Saturninus reminds me of the prophet #Mani (#Manichaeanism), also the #Cathars, with an outer circle of normies with a newly acquired faith or belief-system (religion 101) and inner circle of initiates with #vegetarianism being one of the ethical requirements to qualify for #initiation into the #mysteries of #spiritual practice: https://www.youtube.com/watch?v=RE4LBiSgqRQ&list=PLQl0BipQXVy436zEJKefujiWraA1FSpKQ&index=6&ab_channel=M.DavidLitwa

#Pythagorean #Harp #demo

https://yewtu.be/watch?v=4cUy13R5SC4

Alt. link.:

https://www.youtube.com/embed/4cUy13R5SC4

RT @giohio@twitter.com

#Geometry teachers! Remember this resource when you introduce the #Pythagorean Theorem to your students next year. There are 40 dissection-themed "proofs" of the #Pythagorean Theorem in this @geogebra@twitter.com book: https://ggbm.at/jFFERBdd
#mtbos #iTECHmath @HCESCtech@twitter.com

🐦🔗: https://twitter.com/giohio/status/1137084234596835328

Time for #ProofInAToot! (Well, okay, in two toots.)

Let's generate #Pythagorean triplets \(a^2 + b^2 = c^2\). Dividing by \(c^2\), this is equivalent to finding rational points on the unit circle, \(x^2 + y^2 = 1\).

Consider the line of rational slope \(t\) going through \((-1,0)\), which results in the following system:
\[
\begin{align*}
y &= tx + t\\
x^2 + y^2 &=1
\end{align*}
\]
Its nontrivial solution is
\[
\begin{align*}
x &= \frac{1 - t^2}{1+t^2}\\
y &= \frac{2t}{1+t^2}.
\end{align*}
\]

I just found something interesting about #Pythagorean triples. You can systematically enumerate them in a tree using some magical-looking matrices!

https://en.wikipedia.org/wiki/Tree_of_primitive_Pythagorean_triples

The proof that the matrices work is a boring calculation and gives no insight into how the matrices were derived in the first place. I think I'm going to be reading some papers to figure out where those matrices came from.