in the book Thirty-three Miniatures: Mathematical and Algorithmic Applications of in problem 28 The Secret Agent and the Umbrella page 132 (pdf 140)
we want to find an orthogonal reperesentation of the graph C5 to do this the book says :
To prove our main theorem we will need an interesting orthogonal representation ρLU of the graph C5 in R3, the “Lov´asz umbrella”. Let us imagine a collapsed umbrella with five ribs and the tube made of the vector e1 = (1, 0, 0). Now we open it slowly until the moment when all pairs of non-neighboring ribs become orthogonal:
At this moment, the ribs define unit vectors v1, v2, . . . , v5. When we assign the ith vertex of the graph C5 to the vector vi, we get an orthogonal representation ρLU. It is easy to calculate the opening angle of the umbrella: We obtain angle between vi , e1 is 5^(−1/4)
i want to prove that the angle between vi , e1 is 5^(−1/4). call it $\theta$
i mapped the umbrella to a plane so i got a pentagon since all the ribs have the same angle with the tube then it's a regular pentagon .
if i can find the radius of the circle then i can find the angle since the tube ,radius of the circle and one of the ribs make a right triangle since we know the ribs are of length 1 then $sin(\theta) = r$
to find r consider :
if i find y then $r = y/(2*sin(73))$
to find y i use the fact that we know non adjacent ribs must be orthogonal . so consider the triangle made by two non adjacent ribs and y . since this is also a right triangle then $y = \sqrt2$
so $r = \sqrt2/(2*sin(73))$
so $\theta = sin^{-1}(\sqrt2/(2*sin(73)))$
which gives $\theta = 0.832202$
but 5^(−1/4) = 0.6687403
which are different . what am i doing wrong ?
We have $\sin (2\pi / 5) = \frac{\sqrt{10 + 2\sqrt{5}}}{4}$. Hence, $$ r = \frac{\sqrt{2}}{2} \cdot \frac{1}{\sin (2\pi / 5)} = \frac{2}{\sqrt{5 + \sqrt{5}}}, $$ and $$ r^2 = \frac{4}{5 + \sqrt{5}} = 1 - \frac{1}{\sqrt{5}}, \cos\theta = \sqrt{1 - r^2} = \frac{1}{\sqrt[4]{5}}. $$