## 4D Crinds

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

### 4D Crinds

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 3.1415926535897932384626433832795028841971693int 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 (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
Dionian

Posts: 32
Joined: Tue Nov 27, 2018 2:13 pm
Location: Queens, NY

### Re: 4D Crinds

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 (30.15 KiB) Viewed 92 times
ndl
Dionian

Posts: 32
Joined: Tue Nov 27, 2018 2:13 pm
Location: Queens, NY

### Re: 4D Crinds

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.
Trionian

Posts: 69
Joined: Sat Mar 18, 2017 1:42 pm

### Re: 4D Crinds

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.
ndl
Dionian

Posts: 32
Joined: Tue Nov 27, 2018 2:13 pm
Location: Queens, NY

### Re: 4D Crinds

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!
Marek14
Pentonian

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Joined: Sat Jul 16, 2005 6:40 pm

### Re: 4D Crinds

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?
Trionian

Posts: 69
Joined: Sat Mar 18, 2017 1:42 pm

### Re: 4D Crinds

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.
Marek14
Pentonian

Posts: 1120
Joined: Sat Jul 16, 2005 6:40 pm

### Re: 4D Crinds

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.
ndl
Dionian

Posts: 32
Joined: Tue Nov 27, 2018 2:13 pm
Location: Queens, NY