Clóvis Iarossi

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[mathhombre]

Sangaku Synchronicity.

I love the Japanese sacred geometry known as sangaku. As I was preparing for a class on proof in HS geometry, I was thinking about what could add a level of intrigue to the circle, triangle and parallel line proofs - and thought of sangaku. I became quite taken with the 5 circles in a square one pictured above; beautiful Pythagorean Theorem connections.

I was pretty stumped working it out, so I set it up in GeoGebra to trace the possible squares - and found that gorgeous limaçon hiding! Wow!

So then I had to work up the whole thing in GeoGebra. Serendipity in math is a sweet thing, no?

If you’d like more, Jean-Paul Berroir has some gorgeous Sangaku on GeoGebraTube

[intothecontinuum]

What do you see? Some kind of band spinning around, or something twisting?

The motion of the first band shown above is ambiguous. Its more obvious that the band on the left is spinning, whereas the band on the right is twisting. The particular dynamics are made evident by watching the two black points on the bands edges. But without these defining features the motion of the gray band is indistinguishable between the two cases.

Perhaps the band is not moving at all, and instead its just the observer’s perspective rotating around this twisted yet inert band!

Can you think of any other shape that might exhibit this kind of symmetry?

Inspired by Bees&Bombs

Mathematica code:

x[ r_, w_, s_, t_] := {r (1 + w*Cos[2 Pi*t]) Cos[2 Pi *s], r (1 + w*Cos[2 Pi*t]) Sin[2 Pi*s] , w*Sin[ 2 Pi*t]} P[ Q_, N_, r_, w_, f_, S_, T_] := Table[x[r, w, -s + n/N, f*t + q/Q + f*n/N] , {s, 1/S, 1, 1/S}, {t, 1/T, 1, 1/T}, {q, 0, Q - 1, 1}, {n, 0, N, 1 }] V = P[2, 200, 1, .25, 1, 50, 50]; Manipulate[ Graphics3D[{ {{Thickness[.001], Line[V[[s, t]]]}, Table[ {FaceForm[GrayLevel[.5]], Opacity[1], EdgeForm[None], Polygon[ {V[[s, t, 1, n]], V[[s, t, 2, n]], V[[s, t, 2, n + 1]], V[[s, t, 1, n + 1]], V[[s, t, 1, n]]}]}, {n, 1, 200, d}]}, {PointSize[p], Black, Point[{V[[s, t, 1, 1]], V[[s, t, 2, 1]]}]}}, Background -> White, Lighting -> "Neutral", PlotRange -> 3.5, ViewPoint -> .4 {1, 1, .6}, Boxed -> False, ImageSize -> 500], {t, 1, 50, 1}, {s, 1, 50, 1}, {d, 1, 5, 1}, {p, {0, .015}}]

[deafeningpalacetreehumanoid]

Parabéns grande Amor da minha vida,... Felicidade, saúde, força, coragem e sucesso sempre a você 👏🏽👏🏽👏🏽🥰 Estou em uma busca contínua em ser um humano melhor ao seu lado. Te amo Amor,... Meu Anjo 😍❣️🥰💞 https://www.instagram.com/p/CTv1AvxrPaJuK30lGNqH0u7uTTgH0_DDDMF4OY0/?utm_medium=tumblr

[deafeningpalacetreehumanoid]

Uma boquinha no Jazz Burgers 🤤😋 (em Jazz Burgers) https://www.instagram.com/p/CP_wEHlLpZ-9lYS3jeTfeG_l4I2oS4vJXzDrgg0/?utm_medium=tumblr

[deafeningpalacetreehumanoid]

My Love 😘 https://www.instagram.com/p/CN3EHzFAxZrwEy7VlcfzB5fYioFS-ifa1L_EQo0/?igshid=14wkkv9m37gjd

[deafeningpalacetreehumanoid]

[deafeningpalacetreehumanoid]

#amazonia #serrasul (em Canaã Dos Carajás,brasil) https://www.instagram.com/p/CIwEY0EgCPZ2ntUvh2nd59jvvhBS2ILXzZ99HE0/?igshid=wouvydr82l9y

[deafeningpalacetreehumanoid]

#peixefrito #camarãoempanado #filédepeixe #tucunaré #filédedourada #filédepescada #farrofa #vinagrete #purêdebatata #limão #cocacolagelada #colegasdetrabalho Hoje pode 🤤😋🤭 (em Acarajé do Baiano) https://www.instagram.com/p/CIvsuSPrK3muuyE1ZyDUvYXFzBt24IDh43HRG40/?igshid=y6nlaompdik4

[deafeningpalacetreehumanoid]

🇧🇷 Combustível de domingo . 🇬🇧 Sunday fuel Ouvindo My Hometown (Bruce Springsteen) 👍🏽💪🏽 (em No Mundo) https://www.instagram.com/p/CGxyZjCLTzI/?igshid=fzt4qqhudn43

[deafeningpalacetreehumanoid]

Stapelia grandiflora https://www.instagram.com/p/CGie0fUnioI/?igshid=ev8irob0rv4h

[deafeningpalacetreehumanoid]

Primas em período reprodutivo,... 1° plano (SAPUCAIA) e em 2° plano (CASTANHEIRA) #Natureza #Amazônia (em Em Algum Lugar Do Mundo) https://www.instagram.com/p/CGKYAZxnuI6/?igshid=12puwxrzcry5v

[deafeningpalacetreehumanoid]

E o dia está assim,... 👍🏽💪🏽 (em Canaã Dos Carajás,brasil) https://www.instagram.com/p/CFUiXXQHFJ-6sQbXwG45bgp6i9XRkbGstybsVU0/?igshid=1h3gkgarqlsfv

[deafeningpalacetreehumanoid]

[deafeningpalacetreehumanoid]

Assista a "Stand By Me | Playing For Change Band | Live in Brazil" no YouTube

[deafeningpalacetreehumanoid]

em Canaã dos Carajas https://www.instagram.com/p/CEPmuZJpo54SExZMfHxGPO8E_QW1oC6cG3XDq40/?igshid=c8k5oar03mlm

[deafeningpalacetreehumanoid]

em Chiquinho Sorvetes - Marabá https://www.instagram.com/p/CDvvEr9JrAlmjnggT4u-EvlrHqKbmCAJYa8oco0/?igshid=cp0qhp4ga8b4

[deafeningpalacetreehumanoid]

Excelente 👏🏾👏🏾👏🏾