Random IVM Tetrahedron Volumes (with gmpy2)

According to a proof by Dr. Robert Gray, at one time archived on a Synergetics listserv, any tetrahedron of IVM vertexes (assuming non-coplanar) will have a whole number volume.

In this Notebook, we randomly generate IVM tetrahedrons and compute their volumes. Within the error of floating point, we demonstrate this result.



In [1]:

    
from tetravolume import Qvector



In [2]:

    
import gmpy2
from gmpy2 import mpfr
gmpy2.get_context().precision=200
mpfr(0)









    Out[2]:





mpfr('0.0',200)



In [3]:

    
from itertools import permutations



In [4]:

    
combos = set()
for combo in permutations((0,1,1,2)):
    combos.add(combo)
combos = list(combos)
combos









    Out[4]:





[(0, 1, 1, 2),
 (1, 2, 0, 1),
 (0, 2, 1, 1),
 (0, 1, 2, 1),
 (2, 1, 0, 1),
 (1, 1, 2, 0),
 (1, 2, 1, 0),
 (2, 1, 1, 0),
 (1, 0, 2, 1),
 (2, 0, 1, 1),
 (1, 0, 1, 2),
 (1, 1, 0, 2)]

One frequency:



In [5]:

    
import random

def any_ball():
    # random coefficients for arbitrary choice of vectors...
    coeffs = [random.randint(0, 300) for _ in range(12)]
    vectors = ([coeffs[i] * Qvector(combos[i]) for i in range(12)])
    vector_sum = Qvector((mpfr(0),mpfr(0),mpfr(0),mpfr(0)))
    for v in vectors:
        vector_sum = vector_sum + v
    return vector_sum

any_ball()









    Out[5]:





ivm_vector(a=mpfr('0.0',200), b=mpfr('392.0',200), c=mpfr('222.0',200), d=mpfr('642.0',200))



In [6]:

    
A,B,C,D = any_ball(), any_ball(), any_ball(), any_ball()
A









    Out[6]:





ivm_vector(a=mpfr('492.0',200), b=mpfr('329.0',200), c=mpfr('0.0',200), d=mpfr('267.0',200))



In [7]:

    
A-B









    Out[7]:





ivm_vector(a=mpfr('248.0',200), b=mpfr('263.0',200), c=mpfr('0.0',200), d=mpfr('345.0',200))



In [8]:

    
(A-B).length()









    Out[8]:





mpfr('182.36501857538358499472121928182087681992193154377123883742419',200)



In [9]:

    
lengths = [("AB", (A-B).length()),
           ("AC", (A-C).length()),
           ("AD", (A-D).length()),
           ("BC", (B-C).length()),
           ("CD", (C-D).length()),
           ("DB", (D-B).length())]



In [10]:

    
lengths









    Out[10]:





[('AB',
  mpfr('182.36501857538358499472121928182087681992193154377123883742419',200)),
 ('AC',
  mpfr('600.15164750252914722690451617422631602104978692970170354841761',200)),
 ('AD',
  mpfr('551.34199912576948539719539536015344824528780410498293183661653',200)),
 ('BC',
  mpfr('511.7528700456891428844478216824504677275886445592614561129447',200)),
 ('CD',
  mpfr('598.36109499197890280611882194630845066148913520677993892985788',200)),
 ('DB',
  mpfr('383.2270867253513957959972492873305724441048590574352961993633',200))]



In [11]:

    
from tetravolume import Tetrahedron



In [12]:

    
t = Tetrahedron(*[lengths[i][1] for i in range(6)])



In [13]:

    
t.ivm_volume()









    Out[13]:





mpfr('20009460.0',200)



In [14]:

    
def demo():
    A,B,C,D = any_ball(), any_ball(), any_ball(), any_ball()
    lengths = [("AB", (A-B).length()),
           ("AC", (A-C).length()),
           ("AD", (A-D).length()),
           ("BC", (B-C).length()),
           ("CD", (C-D).length()),
           ("DB", (D-B).length())]
    t = Tetrahedron(*[lengths[i][1] for i in range(6)])
    return t.ivm_volume()

demo()









    Out[14]:





mpfr('16319836.0',200)



In [15]:

    
t.xyz_volume()









    Out[15]:





mpfr('18865099.805174635666877858142242985195480287075042724609375',200)



In [16]:

    
from IPython.display import YouTubeVideo
YouTubeVideo("wLHescQT2ZI") # https://youtu.be/wLHescQT2ZI









    Out[16]: