4D Crinds

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

4D Crinds

Postby ndl » Fri Feb 08, 2019 6:54 am

I wrote a little program in c to calculate vertices to approximate the cylindrical crind and create a .off file:

Code: Select all
#include <stdio.h>
#include <math.h>
#define PI 3.1415926535897932384626433832795028841971693

int main ()
{
  int i;
  int realm;
  double points[31];
  int xval = 0;
  int zval = 30;
  double x = 0, y = 0, z = 0, w = -1;
  double change;
  FILE *output;
  output = fopen("cc.off", "w");
  fprintf (output, "4OFF\n14162 0 0 0\n");
  for (i = 0; i <= 30; i++)
    {
      points[i] = sin(PI*i/60);
    }
  fprintf(output, "% .16f % .16f % .16f % .16f\n", x, y, z, w);
  for (realm = -29; realm <=29; realm++)
    {
      change = 1 - cos(PI*realm/60);
      for (i = 0; i <= 239; i++)
        {
          if ((i/60) % 2 == 0)
            {
              x = change*points[xval];
            }
          else
            {
              x = change*-1*points[xval];
            }
          if (i < 120)
            {
              y = change;
            }
          else
            {
              y = change*-1;
            }
          if (((i+30)/60) % 2 == 0)
            {
              z = change*points[zval];
            }
          else
            {
              z = change*-1*points[zval];
            }
          w = sin(PI*realm/60);
          fprintf(output, "% .16f % .16f % .16f % .16f\n", x, y, z, w);
          if ((i/30) % 2 == 0)
          {
            xval++;
            zval--;
          }
          else
          {
            xval--;
            zval++;
          }
        }
    }
  fprintf(output, " 0.0000000000000000  0.0000000000000000  0.0000000000000000  1.0000000000000000\n");
  fclose (output);
  return 0;
}


I loaded it in Stella4d and it calculated the convex hull fo me. Here's a projection of it:

CC.JPG
CC.JPG (39.78 KiB) Viewed 99 times


While playing around with it I discovered it has a perfect sphere cross-section in one orientation!
Last edited by ndl on Sun Feb 10, 2019 6:20 am, edited 1 time in total.
ndl
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Location: Queens, NY

Re: 4D Crinds

Postby ndl » Sun Feb 10, 2019 6:19 am

After playing around with all the types of crinds (bracketopes excluding the tegum sums) I think I discovered a new 4D crind! I don't know how to algebraically define it, but it's like a duocylinder that's been cut by another cylinder in one orientation and not the other, so you are left with cross sections of line-cylinder-line along 2 axes and circle-crind-circle along the other two. Is anyone familiar with this shape?

Here's a render:

New Crind.JPG
New Crind.JPG (30.15 KiB) Viewed 92 times
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Location: Queens, NY

Re: 4D Crinds

Postby username5243 » Sun Feb 10, 2019 11:29 am

ndl wrote:After playing around with all the types of crinds (bracketopes excluding the tegum sums) I think I discovered a new 4D crind! I don't know how to algebraically define it, but it's like a duocylinder that's been cut by another cylinder in one orientation and not the other, so you are left with cross sections of line-cylinder-line along 2 axes and circle-crind-circle along the other two. Is anyone familiar with this shape?

Here's a render:

New Crind.JPG


I think this may be what Marek14 calls a "longdome" - at least the description seems to match. He's posted about it here on several occasions.
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Re: 4D Crinds

Postby ndl » Sun Feb 10, 2019 7:05 pm

Yes! That is what I was trying to describe, thank you. There is a lot of stuff in this forum to sift through I don't know how anyone finds anything.
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Location: Queens, NY

Re: 4D Crinds

Postby Marek14 » Sat Feb 16, 2019 11:10 am

username5243 wrote:
ndl wrote:After playing around with all the types of crinds (bracketopes excluding the tegum sums) I think I discovered a new 4D crind! I don't know how to algebraically define it, but it's like a duocylinder that's been cut by another cylinder in one orientation and not the other, so you are left with cross sections of line-cylinder-line along 2 axes and circle-crind-circle along the other two. Is anyone familiar with this shape?

Here's a render:

New Crind.JPG


I think this may be what Marek14 calls a "longdome" - at least the description seems to match. He's posted about it here on several occasions.


I'm glad that concept is getting attention again :) As someone pointed out in the thread, it has been 14 years!
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Re: 4D Crinds

Postby username5243 » Sat Feb 16, 2019 2:11 pm

One thing I've been thinking: 3D crind is known to be the intersection of two perpendicular cylinders. Can any of the 4D crinds be derived in the similar way, e.g., as the intersection of perpendicular spherinders?
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Re: 4D Crinds

Postby Marek14 » Sat Feb 16, 2019 6:06 pm

username5243 wrote:One thing I've been thinking: 3D crind is known to be the intersection of two perpendicular cylinders. Can any of the 4D crinds be derived in the similar way, e.g., as the intersection of perpendicular spherinders?


Maybe. The maximum function plays a major role in these shapes and that can pretty much work as intersection or union of two shapes symbolized by other expressions.
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Re: 4D Crinds

Postby ndl » Sun Feb 17, 2019 1:20 am

username5243 wrote:One thing I've been thinking: 3D crind is known to be the intersection of two perpendicular cylinders. Can any of the 4D crinds be derived in the similar way, e.g., as the intersection of perpendicular spherinders?


Sure, I think that would make the Cylindrical Crind.
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Location: Queens, NY


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