Random IVM Tetrahedron Volumes

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]:

    
from itertools import permutations



In [3]:

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









    Out[3]:





[(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 [4]:

    
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((0,0,0,0))
    for v in vectors:
        vector_sum = vector_sum + v
    return vector_sum

any_ball()









    Out[4]:





ivm_vector(a=365, b=159, c=336, d=0)



In [5]:

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



In [6]:

    
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 [7]:

    
lengths









    Out[7]:





[('AB', 373.0321701944753),
 ('AC', 333.39616074574104),
 ('AD', 476.14808620848197),
 ('BC', 534.9635501601955),
 ('CD', 544.9155897935019),
 ('DB', 280.3194606159194)]



In [8]:

    
from tetravolume import Tetrahedron



In [9]:

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



In [10]:

    
t.ivm_volume()









    Out[10]:





44190451.00000004



In [11]:

    
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[11]:





14170140.999999965



In [12]:

    
t.xyz_volume()









    Out[12]:





41663156.754389174



In [13]:

    
from IPython.display import YouTubeVideo
YouTubeVideo("v8vjw3gCBgs") # https://youtu.be/v8vjw3gCBgs









    Out[13]: