Pursuing the Perfect Parallelepiped Auxiliary Materials

by Clifford A .  Reiter and Jordan O .  Tirrell

Theorem 1 Proof Material

len2[v_] := v . v

pvn[p_, q_, n_] := {(p^2 + q^2 - n^2)/(n), 2  p, 
        2  q}

pvn[p, q, n]

Factor[len2[pvn[p, 0, n]]]

Factor[len2[pvn[p, q, n]]]

Factor[
      len2[{(p^2 + q^2 + r^2 - n^2)/(n), 2  p, 2  q, 
          2  r}]]

Factor[
      len2[{((2  p + 1)^2 + (2  q + 1)^2 + (2  r + 1)^2 - 
                n^2)/(2  n), 2  p + 1, 2  q + 1, 2  r + 1}]]

Factor[((2  p + 1)^2 + (2  q + 1)^2 + (2  r + 1)^2 + 
            n^2)/(2  n)]

Theorem 2 Proof Material

J = {{0, 1, 1, 1}, {1, 0, 1, 1}, {1, 1, 0, 1}, {1, 1, 1, 2}}

pvnx[p_, q_, n_] := {(p^2 + q^2 - n^2)/(n), 2  p, 
        2  q, (p^2 + q^2 + n^2)/n}

pvnx[p, q, n]

J  .  pvnx[p, q, n]

Drop[J  .  pvnx[p, q, n], -1]

Factor[len2[Drop[J  .  pvnx[p, q, n], -1]]]

pvnx2[p_, q_, n_] := {(p^2 + q^2 - n^2)/(n), 2  p, 
        2  q, -(p^2 + q^2 + n^2)/n}

Factor[len2[Drop[J  .  pvnx2[p, q, n], -1]]]

Proposition 3 Proof Material

J = {{0, 1, 1, 1}, {1, 0, 1, 1}, {1, 1, 0, 1}, {1, 1, 1, 2}}

J1 = {{0, 1, 1, 1}, {-1, 0, 1, 1}, {-1, 1, 0, 1}, {-1, 1, 
          1, 2}}

J2 = {{0, -1, 1, 1}, {1, 0, 1, 1}, {1, -1, 0, 1}, {1, -1, 
          1, 2}}

J3 = {{0, 1, -1, 1}, {1, 0, -1, 1}, {1, 1, 0, 1}, {1, 
          1, -1, 2}}

J23 = {{0, -1, -1, 1}, {1, 0, -1, 1}, {1, -1, 0, 
          1}, {1, -1, -1, 2}}

J12 = {{0, -1, 1, 1}, {-1, 0, 1, 1}, {-1, -1, 0, 
          1}, {-1, -1, 1, 2}}

J13 = {{0, 1, -1, 1}, {-1, 0, -1, 1}, {-1, 1, 0, 
          1}, {-1, 1, -1, 2}}

J123 = {{0, -1, -1, 1}, {-1, 0, -1, 
          1}, {-1, -1, 0, 1}, {-1, -1, -1, 2}}

Inverse[J]

J123  .  J123

J12 . J12 . J12 . J12

J13 . J13 . J13 . J13

J23 . J23 . J23 . J23

J . J . J - 3  J . J - 3  J + IdentityMatrix[4]

Theorem 4 Proof Material

Factor[len2[J1 . pvnx[p, q, n]] - len2[pvnx[p, q, n]]]

Expand[3/2  n^2 + (n/2 + p)^2 + (n/2 + q)^2]

Factor[len2[J12 . pvnx[p, q, n]] - len2[pvnx[p, q, n]]]

Expand[3/2  n^2 + (n/2 - p)^2 + (n/2 + q)^2]

Factor[len2[J13  . pvnx[p, q, n]] - len2[pvnx[p, q, n]]]

Expand[3/2  n^2 + (n/2 + p)^2 + (n/2 - q)^2]

Factor[
      len2[J123 . pvnx[p, q, n]] - 
        len2[pvnx[p, q, n]]]

Expand[3/2  n^2 + (n/2 - p)^2 + (n/2 - q)^2]

Proposition 5 Proof Material

Prop5u[p_, q_] := {4  q^2 (p^2 + q^2 - 1), 8  p q^2, 
        8  q^3}

Prop5v[p_, q_, r_] := {4 (r^2 - 1) q^2 + (2 p r + 1)^2, 
        8 q^2 r, -4 q (2 p r + 1)}

Factor[len2[Prop5u[p, q]]]

Factor[len2[Prop5v[p, q, r]]]

Factor[len2[Prop5u[p, q] + Prop5v[p, q, r]]]

Det[{Prop5u[p, q], Prop5v[p, q, r1], Prop5v[p, q, r1]}]

Proposition 6 Proof Material

u := {25 (p^2) - 25 (q^2), 50 p q, 0}
v := {25 (p^2) - 25 (q^2), -50 p q, 0}
w := {0, -28 p q, 96 p q}

Factor[len2[u]]

Factor[len2[v]]

Factor[len2[w]]

Factor[len2[u + v + w]]

Factor[len2[u - v + w]]

Factor[len2[u - v]]

Factor[len2[u + v]]

Factor[len2[u + v - w]]

Factor[len2[u - v - w]]

Factor[len2[u + w]]

Factor[len2[v - w]]

Factor[len2[u - w]]

Factor[len2[v + w]]