Families of Nearly Perfect Parallelepipeds

Auxiliary Materials

by Daniel S D ' Argenio and Clifford A . Reiter

August 2007

Gives the length squared in factored form :

In[1]:=

len2[X_] := Factor[X . X]

Gives extension in the style of Theorem 1 :

In[2]:=

ext[u_, k_] := Append[2 k u, len2[u] - k^2]

Corollary 2 Extension Checked :

In[3]:=

ext[{a}, b]

Out[3]=

{2 a b, a^2 - b^2}

In[4]:=

len2[ext[{a}, b]]

Out[4]=

(a^2 + b^2)^2

Corollary 3 Extension Checked :

In[5]:=

z3 = ext[ext[{a}, b], c]

Out[5]=

{4 a b c, 2 (a^2 - b^2) c, (a^2 + b^2)^2 - c^2}

In[6]:=

len2[z3]

Out[6]=

(a^4 + 2 b^2 a^2 + b^4 + c^2)^2

Proposition 4 Checked :

In[7]:=

In[8]:=

{u, v, w} = z3 * IdentityMatrix[3]

Out[8]=

( {{4 a b c, 0, 0}, {0, 2 (a^2 - b^2) c, 0}, {0, 0, (a^2 + b^2)^2 - c^2}} )

All squares except two distinct face diagonal conditions

In[9]:=

len2a[u, v, w]

Out[9]=

Proposition 5 Checked :

In[10]:=

u = {1, 1, 0} * z3

Out[10]=

{4 a b c, 2 (a^2 - b^2) c, 0}

In[11]:=

v = {1, -1, 0} * z3

Out[11]=

{4 a b c, -2 (a^2 - b^2) c, 0}

In[12]:=

w = {0, 0, 1} * z3

Out[12]=

{0, 0, (a^2 + b^2)^2 - c^2}

All squares except two distinct body diagonals

In[13]:=

len2a[u, v, w]

Out[13]=

Proposition 6 Checked :

In[14]:=

{p, q} = ext[{a}, b]

Out[14]=

{2 a b, a^2 - b^2}

In[15]:=

{r, s} = ext[{c}, d]

Out[15]=

{2 c d, c^2 - d^2}

In[16]:=

u = (r^2 + s^2) {p, q, 0}

Out[16]=

{2 a b (4 c^2 d^2 + (c^2 - d^2)^2), (a^2 - b^2) (4 c^2 d^2 + (c^2 - d^2)^2), 0}

In[17]:=

v = (r^2 + s^2) {-p, q, 0}

Out[17]=

{-2 a b (4 c^2 d^2 + (c^2 - d^2)^2), (a^2 - b^2) (4 c^2 d^2 + (c^2 - d^2)^2), 0}

In[18]:=

w = 2p {s^2 - r^2, 0, 2 r s}

Out[18]=

{4 a b ((c^2 - d^2)^2 - 4 c^2 d^2), 0, 16 a b c d (c^2 - d^2)}

All squares except two distinct face diagonal conditions

In[19]:=

len2a[u, v, w]

Out[19]=

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