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Posts tagged with pythagorean theorem

OK, not every day. But whenever I shop for packaged retail goods like a coffee or in the grocers.

The Pythagorean theorem demonstrates that a slightly larger circle has twice as much area as a slightly smaller circle.

Pythagorean Theorem  This is how I first really understood the Pythagorean Theorem.  The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.  Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.  Ideas involved here:  scaling properties of squared quantities(gravitational force, skin, paint, loudness, brightness)  circumcircle & incircle  2  This is also how I first really understood 2, now my favourite number.

(Since the diagonal of that square is √2 long relative to the "1" of the interior radius=leg of the right triangle. So the outer radius=hypotenuse=√2, and √2 squared is 2.)

image

And some of us know from Volume Integrals in calculus class that a cylinder's volume = circle area × height — and something like a sausage with a fat middle, or a cup with a wider mouth than base, can be thought of as a “stack” of circle areas
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or in the case of a tapered glass, a “rectangle minus triangle” (when the circle is collapsed so just looking at base-versus-height “camera straight ahead on the table” view).

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The shell-or-washer-method volume integral lessons were, I think, supposed to teach about symbolic manipulation, but I got a sense of what shapes turn out to be big or small volume as well.

http://2.bp.blogspot.com/__wa77chrZVg/SuRA4fj-l8I/AAAAAAAADHM/quRNFMVeHmk/s400/Chou_pei.jpg

By integrating dheight sized slices of circles that make up a larger 3-D shape, I can apply the inverse-square lesson of the Pythagorean theorem to how real-life “cylinders” or “cylinder-like things” will compare in volume.

  • A regulation Ultimate Frisbee can hold 6 beers. (It’s flat/short, but really wide)
    File:Frisbee Catch- Fcb981.jpg
    image
  • The “large” size may not look much bigger but its volume can in fact be.
    image
  • Starbucks keeps the base of their Large cups small, I think, to make the large size look noticeably larger (since we apparently perceive the height difference better than the circle difference). (Maybe also so they fit in cup holders in cars.)




"The Chinese Proof" of the Pythagorean theorem (the little orange square is a², the medium orange square is b², and the large orange square is c²).
Harald Hanche-Olsen:

The righthand picture above appears in the Chou pei suan ching 周髀算經 (ca. 1100 B.C.), for the special (3,4,5) pythagorean triple….
…the earliest known proof of Pythagoras is given by Zhoubi suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) (c. 100 B.C.E.-c. 100 C.E.)
…
[T]his proof, with the exclamation `Behold!’, is due to the Indian mathematician Bhaskara II (approx. 1114-1185) …
…
Jöran Friberg … presented convincing evidence that the … Babylonians were aware of the Pythagoras theorem around 1800 B.C.E.



Online Zhoubi Suanjing:
卷上:
句股圓方圖:右圖:左圖:

"The Chinese Proof" of the Pythagorean theorem (the little orange square is , the medium orange square is , and the large orange square is ).

Harald Hanche-Olsen:

The righthand picture above appears in the Chou pei suan ching 周髀算經 (ca. 1100 B.C.), for the special (3,4,5) pythagorean triple….

…the earliest known proof of Pythagoras is given by Zhoubi suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) (c. 100 B.C.E.-c. 100 C.E.)

[T]his proof, with the exclamation `Behold!’, is due to the Indian mathematician Bhaskara II (approx. 1114-1185) …

Jöran Friberg … presented convincing evidence that the … Babylonians were aware of the Pythagoras theorem around 1800 B.C.E.

http://library.thinkquest.org/05aug/01951/images/content/china_proof.jpg

http://2.bp.blogspot.com/__wa77chrZVg/SuRA4fj-l8I/AAAAAAAADHM/quRNFMVeHmk/s400/Chou_pei.jpg

Online Zhoubi Suanjing:

卷上:
句股圓方圖:右圖:左圖:




Pythagorean Theorem
This is how I first really understood the Pythagorean Theorem.
The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.
Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.
Ideas involved here:
scaling properties of squared quantities(gravitational force, skin, paint, loudness, brightness)
circumcircle & incircle
√2
This is also how I first really understood √2, now my favourite number.

Pythagorean Theorem

This is how I first really understood the Pythagorean Theorem.

The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.

Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.

Ideas involved here:

  • scaling properties of squared quantities
    (gravitational force, skin, paint, loudness, brightness)
  • circumcircle & incircle
  • √2

This is also how I first really understood √2, now my favourite number.


hi-res




Pythagoras
visual proofs of the Pythagorean Theorem, a² + b² = c²

Pythagoras

visual proofs of the Pythagorean Theorem, a² + b² = c²


proof 3

proof 2

proof 4


hi-res