OT: Möbius strip [NCA / SHAM]

posted by Helmut Homepage – Vienna, Austria, 2020-06-24 11:29 (95 d 10:09 ago) – Posting: # 21572
Views: 1,690

Hi Nastia,

» Hey, how did you know that I've got a model of it? :-D

I didn’t know but I’m not surprised.

» Unfortunatelly I am not good at topology…

cup → bagel → cup → …
[image]
© Lucas Vieira Barbosa

Welcome to the club. The worst is knot theory – I have books about it. Headaches guaranteed.
For sure you know what happens if we cut a Möbius strip once. Do you know what happens if we cut it twice? Makes a great party joke.

» But I know the answers for the questions what do we need to drill a square hole…

Punch, OK. But drill?

» … and also what profile one should use to ride a bike with square wheels ;-)

So do I.


https://www.r-bloggers.com/topological-tomfoolery-in-r-plotting-a-mobius-strip
library(rgl)
library(plot3D)
###############
# Möbius band #
###############

R <- 5
u <- seq(0, 2 * pi, length.out = 100)
v <- seq(-1, 1, length.out = 100)
m <- mesh(u, v)
u <- m$x
v <- m$y
x <- (R + v/2 * cos(u /2)) * cos(u)
y <- (R + v/2 * cos(u /2)) * sin(u)
z <- v/2 * sin(u / 2) 
invisible(open3d())
bg3d(color = "#F3F3F3")
surface3d(x, y, z, color= "#87CEFA")
title3d(main = "Moebius band")
# Rotate with mouse-left, wheel to zoom.
################
# Klein bottle #
################

u <- seq(0, pi, length.out = 100)
v <- seq(0, 2 * pi, length.out = 100)
m <- mesh(u, v)
u <- m$x
v <- m$y
x <- (-2 / 15) * cos(u) * (3 * cos(v) - 30 * sin(u) + 90 * cos(u)^4 * sin(u) -
      60 * cos(u)^6 * sin(u) + 5 * cos(u) * cos(v) * sin(u))
y <- (-1 / 15) * sin(u) * (3 * cos(v) - 3 * cos(u)^2 * cos(v) - 48 *
      cos(u)^4 * cos(v) + 48 * cos(u)^6 * cos(v) - 60 * sin(u) + 5 * cos(u) *
      cos(v) * sin(u) - 5 * cos(u)^3 * cos(v) * sin(u) - 80 * cos(u)^5 *
      cos(v) * sin(u) + 80 * cos(u)^7 * cos(v) * sin(u))
z <- (+2 / 15) * (3 + 5 * cos(u) * sin(u)) * sin(v)
invisible(open3d())
bg3d(color = "#F3F3F3")
surface3d(x, y, z, color= "#87CEFA", alpha = 0.5)
title3d(main = "Klein bottle")

Dif-tor heh smusma 🖖
Helmut Schütz
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