Mathematics Is Fun

https://gist.github.com/tthtlc/3ee685f66a6addc4082fde0a65b860fa

And instead of a circle, which is drawn as a polygon of 30 sides, we can fixed it to TWO-gon:

The rod-shaped structure formed a flat surface - with top and bottom surface.

How to make it into a Mobius surface? So what if now the generator of the 2gon is shifted by a angle offset as it goes round the circle - and the total angular offset should total 180 degrees?

This is the result:

And the source code is here:

https://gist.github.com/tthtlc/9db1b5a9609932a75f70f0d347b62228

Instead of a 2-gon, why not change it to a 3-gon (triangle) and you will get this:

Instead of triangle (ngon=3) or circle earlier (ngon=30), what if we want to form helix? Then instead of drawing circle by itself, we need to rotate the circle while drawing the circle, forming this:

And the source code is here:

https://gist.github.com/tthtlc/43294d2f4346d5da834ad6aa6278dd13

And what if you make the circles move in a sinusoidal pattern around at the periodic frequency of 2?

The above is the result:

https://gist.github.com/tthtlc/5cc6268e7e8aa0f0025405557951f890

And in case you think the single colored rings are too boring:

https://gist.github.com/tthtlc/7245d5b53513fe13966eca1758921826

https://gist.github.com/tthtlc/d53c71446e58d9b09978aa1fc7e94e76

Jun 29, 2019

Mobius Band (variations)

https://gist.github.com/tthtlc/df6b0cdf017c89dfa7bbc78c1b72219c

Rotating triangles around a hexagon

https://gist.github.com/tthtlc/a39d0f79d8013b102905a14973f28a1d

Jun 28, 2019

Creating a torus (animated and variations)

Looking at the Pentagon torus:

And its source code:

https://gist.github.com/tthtlc/1e1618d0b970699abb06780b42e116b7#file-pentagon_torus-py

And changing the rings to 30:

https://gist.github.com/tthtlc/1e1618d0b970699abb06780b42e116b7#file-pentagon_torusv2-py

And if you use triangle surface to create the Pentagonal torus in 3D:

https://gist.github.com/tthtlc/c67b54382ce34bd11ec2d402d6ddea1d

What are the difference in programming? In the previous two, the torus is created by a prebuilt function called (look into the "display()" function for how the object is generated) glutWireTorus().

global xrot, yrot, zrot
global ndisc
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT)
glLoadIdentity()
gluLookAt(
0.0, 0.0, 10.0,
0.0, 0.0, 0.0,
0.0, 1.0, 0.0)
glRotatef(xrot, 1.0, 0.0, 0.0)
glRotatef(yrot, 0.0, 1.0, 0.0)
glRotatef(zrot, 0.0, 0.0, 1.0)
glColor3f(0.5, 0.0, 1.0)
glutWireTorus(0.5,1.5,5,5)
glFlush()
glutSwapBuffers()

But in the case of triangulated surface generated torus it is built from scratch using "triangle", and thus can be controlled via use of cosine() and sine() to generate the triangles for the surface of the torus. The technique is to generate a list of coordinates of the vertices of the surface, and the normal vector centering on that coordinate.

vertices = []
normals = []
u_step = 2 * pi / (slices)
v_step = 2 * pi / (inner_slices)
u = 0.
for i in range(slices+1):
cos_u = cos(1*u)
sin_u = sin(1*u)
v = 0.
for j in range(inner_slices):
cos_v = cos(1*v)
sin_v = sin(1*v)
d = (radius + inner_radius * cos_v)
x = d * (cos_u +2)* cos_u
y = d * (cos_u +2)* sin_u
z = inner_radius * sin_v
nx = cos_u * cos_v
ny = sin_u * cos_v
nz = sin_v
vertices.extend([x, y, z])
normals.extend([nx, ny, nz])
v += v_step
u += u_step
# Create a list of triangle indices.
indices = []
for i in range(slices):
for j in range(inner_slices):
p = i * inner_slices + j
indices.extend([p, p + inner_slices, p + inner_slices + 1])
indices.extend([p, p + inner_slices + 1, p + 1])
self.vertex_list = batch.add_indexed(len(vertices)//3,
GL_TRIANGLE_STRIP,
group,
indices,
('v3f/static', vertices),
('n3f/static', normals))

Jun 27, 2019

The Einstein Podolsky Rosen (EPR) Paradox - A simple explanation

https://www.youtube.com/watch?v=0x9AgZASQ4k

https://www.youtube.com/watch?v=5HJK5tQIT4A

https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf

https://www.khanacademy.org/partner-content/bjc/2015-challenge/2015-physics/v/bjc-epr-paradox