I see myself in Alexandria of Egypt instead of Euclid while teaches his disciples a lesson in the fourth century b.c.

I’d like to share some steps with you:

The postulate 16 demonstrates how to inscribe an icosahedron inside a sphere by constructing two regular pentagons inscribed within the relative circles and rotated among them by 36 degrees.

These pentagons then form two pentagonal pyramids that connect the sides of the base to the related vertex.

At the end of this long and accurate demonstration Euclide concludes as follows with the final “Corollary”.

Ἐκ δὴ τούτου φανερόν, ancient greek

By virtue of what I have shown above (ie the demonstration of how an icosahedron is inscribed in the sphere).

ὅτι ἡ τῆς σφαίρας διάμετρος δυνάμει πενταπλασίων ἐστὶ τῆς ἐκ τοῦ κέντρου τοῦ κύκλου, ἀφ’ οὗ τὸ εἰκοσάεδρον ἀναγέγραπται, ancient greek

I can state that the diameter of the sphere at the square is five times the distance from the center of the circle used to build the icosahedron,

(Explanation: Let us think of two opposing vertices one in the upper pentagon and the other in the lower pentagon of the icosahedron. Being opposite vertices their distance is equal to the diameter of the sphere. If we trace a segment that goes from the top vertex up to plane of the pentagon below, this segment as shown by Euclid, is equal to the radius of the circle in which the petagon is inscribed. Then we connect this point to the opposite vertex, this distance is equal to the diameter of the circle in which pentagon is inscribed.

Since the triangle thus formed is rectangular we will have the hypotenuse (diameter of the sphere) squared to be the square of the radius of the circle in which the pentagon is inscribed plus the square of the diameter of the circle ie 4 squares of the radius for a total of 5 squares of the radius. Euclid continues as follows:)

καὶ ὅτι ἡ τῆς σφαίρας διάμετρος σύγκειται ἔκ τε τῆς τοῦ ἑξαγώνου καὶ δύο τῶν τοῦ δεκαγώνου τῶν εἰς τὸν αὐτὸν κύκλον ἐγγραφομένων. ancient greek

I also state that the diameter of the sphere corresponds to one side of a hexagon plus two sides of a decagon, inscribed in the same circle.

(Explanation: think of the two equal pentagonal pyramids and rotated 36 degrees, their bases are separated by a distance equal to the radius of the circle in which the pentagon is inscribed (ie the side of inscribed hexagon), the pyramids instead have for height the side of the decagon. We know today that the side of a decagon is equal to 1 / golden ratio.

Now 2 * (1 / golden section) = sqrt (5) -1, let’s add 1 we get sqrt (5). In his Elements Euclid has never used a formula (the formulas will be introduced more than a thousand years later) but concepts well known to his disciples as the sides of the hexagon and the decagon.

Eventually Euclid says:

ὅπερ ἔδει δεῖξαι

QED (quod erat demonstrandum)

what I was trying to demonstrate.