Problem 3

For each

k=1,2,3,4,5

, find necessary and sufficient conditions on

a>0

for the existence of a tetrahedron with

k

edges of length

a

and the remaining

6-k

edges of length

1

Step 4 of 5: Construct three long edges

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In plain words

For three special edges, choose whether they form the small face or the three edges from the apex; the two constructions complement one another.

k=3:\quad 0<a<\sqrt3\ \text{or}\ a>\frac1{\sqrt3};\qquad (0,\infty)=(0,\sqrt3)\cup\left(\frac1{\sqrt3},\infty\right)

Detailed analysis

For $a\le1$ , let the three $a$ -edges form one face; the fourth vertex can be chosen above that face precisely while $a<\sqrt3$ . For $a\ge1$ , let the three unit edges form a face and join its vertices to the fourth vertex by $a$ -edges; this construction works when $a>1/\sqrt3$ . These overlapping ranges cover every $a>0$ (with $a=1$ giving the regular tetrahedron).