Pythagorean Triples (and Quadruples) Are Everywhere!

In this episode, I’ll show some cool ways “Pythagorean triples” are hiding all over geometry, and then discuss the underrated “Pythagorean quadruples”!

This episode about Pythagorean triples/quadruples is the first part of a 2-episode arc. Next episode (coming in a week) will continue this topic into Fermat's Last Theorem and beyond. I love you all, thanks for watching. Also make sure to check out my @Domotro channel if you want to see my bonus content like livestreams (and other links in this video description).

Leave a comment with your favorite animal cameo in this video (there's a squirrel carrying a nut, some cute cats, and some cool bird noises).

This was filmed by Carlo Trappenberg.
Special thanks to Evan Clark and to all of my Patreon supporters:
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Domotro
1442 A Walnut Street, Box # 401
Berkeley, CA 94709

If you want to try to help with Combo Class in some way, or collaborate in some form, reach out at combouniversity(at)gmail(dot)com

In case people search any of these terms, some of the topics discussed in this episode are: Pythagorean triples, Pythagorean quadruples (and Pythagorean quintuples and beyond), the Pythagorean formula (a squared plus b squared equals c squared), right triangles, circles and spheres, squares and rectangles, gridpoints with integer coordinates on a coordinate plane, points on the unit circle with rational coordinates, other geometric analogies, and more. (There's also some cool cats and squirrels in this video haha).

DISCLAIMER: Do not copy any uses of fire, sharp items, or other dangerous tools or activities you may see in this series. These videos are for educational (and entertainment) purposes.

[ComboClass]

This episode about Pythagorean triples/quadruples is the first part of a 2-episode arc. Next episode (coming in a week) will continue this topic into Fermat's Last Theorem and beyond. I love you all, thanks for watching. Also make sure to check out my @Domotro channel if you want to see my bonus content like livestreams (and other links in this video description).

[me]

Let's play a game. Take a shot every time he drops something

[sajrra]

I guess I found a way to generate Pythagorean Triplets when I found out that x²+x+(x+1)=(x+1)². Like 12² +12+13 =13².
Take two consequtive numbers which add to a Square Number like 60+61=11² and you got a solution [60²+11²=61²]
If you want a triplet with the biggest number NOT being consequtive, search for 4x+4=y² or 6x+9=y² (6*12+9=9² works therefore we Re-found 12²+9²=15²)
The following equations to solve for a square-number are 6x+16 ; 8x+25, 10x+36, 12x+49.. n o t i c e the pattern?

[speculativebubble]

Amazing content, as always, thank you!

[mismis]

An easy way I found to remember the formula for triples is to square x+iy
(x+iy)^2 = x^2-y^2 + 2ixy
The modulus, if you care to calculate it is x^2+y^2.
Graphically it is easy to see that for any x, y :
(x^2-y^2)^2 + (2xy)^2 = (x^2+y^2)^2.
There is the formula

[marble]

Pythogorean triples teached in grade -2
Great video

[davidshechtman]

Every odd number that is also a square is guaranteed to be the next odd number in a sequence of odds all added together to make the next smaller square so that when said odd square is added to the perimeter of the next smaller square it becomes a larger square formed from the sum of two smaller ones. The same technique applied to a given square summed with a *partial sum of odds* - (13 + 15 + 17 + 19 = 64, say) is more challenging to characterize.

[murphthegangster]

A always loved finding as many Pythagorean triples as I could, glad you covered this lol

[mienzillaz]

Mathologer had nice video on how to generate such trees.

[zeek]

Wow! This is really unique stuff that must have taken a long time to put together. Thank you!

[MarloTheBlueberry]

Thank you, Combo Class. You make education fun, and you also make jokes from time to time, unlike some teachers who just go, "And then blah is equal to b+a+h-g³/√π÷5.4" . Keep doing rhe great work, and you will be one of the top Education Channels! Remember me when you get to the top.

[publiconions]

Whoa -- so i guess that means there is an infinite amount of rational x, y points on the unit circle?... that blows my mind! My naive intuition would've been that there were infinite irrational, but finite rational. Super cool!

[garymemetoo]

Informative, entertaining, and insightful as usual. The squirrels are always a nice touch. And the cats, too.

[isaacthek]

Each odd number a is part of a unique Pythagorean triple where b is one less than c.

[iamwhatitorture]

For Pythagorean quintuples, wouldn't a diagonal on a whole number hypercube be always be a whole number?
Like it's just the square root of 4*a², so 2a, meaning you'd have a primitive quintuple that's 1² +1² + 1² + 1² = 2². Honestly that's cool

[the_eternal_student]

I have been wanting to see an example of sq rt 5.

[markzambelli]

A few years ago I had my mind blown finding out that 500 years before Pythagoras was even born, Egyptian children were doing maths homework (on clay tablets) that involved many different 'Pythagorean' Triples... it's the main reason I started eating Broad (fava) beans again and peeing at right-angles to the wind.

[bozhidarmihaylov]

Beginning with rationally drawn Coordinate plane with length of 3 and a bit units 😂 👍

[AloisMahdal]

Is it true that for each point on the plane with integer coordinates x and y, say, x=3, y=7, there's an equivalent Pythagorean triple that would have rational coordinates but sit on the unit circle? I.e. there's a solution where x/y = a/b and a^2 + b^2 = 1, right? If that is true, and we could find all these (infinite) pairs and make a "rational circle", a "sparse" circle that would only consist of rational solutions to a^b + b^c? If so, how would that circle look like? Would density of points on that rational circle uniform?

[aashsyed]

Please do a video about integer solutions to a²+b²+c²=d²