As we saw last week, the centers of circles of radii 1, 2, and 3 tangent to each other are the vertices of a right triangle. And not just any one, but the one with sides 3, 4 and 5, none other than the sacred triangle of the Egyptians, who knew that the angle opposite the longest side of this triangle was right, although it is not certain that they generalized this result to all the triangles whose sides satisfy the relationship a² = b² + c² (that is, they knew the Pythagorean theorem).
As Salva Fuster points out: “For the circles tangent to each other with radii 1, 2 and 3, once their centers have been drawn forming a right triangle with sides 3, 4 and 5, it is easy to see that the radius of the outer tangent circle to those three It will be 6, and that its center will be found precisely at the point that would form a rectangle with the other three centers.” (Because?).
Finding the radius of the interior tangent circle by geometric methods is not so simple. We can resort to the formula:
Q² + R² + S² + T² = 1/2 (Q + R+ S + T)²
But, as we saw, the calculations are long and cumbersome, so it is convenient to use another formula that gives us T directly:
T = Q + R + S + 2√(QR + QS + RS)
T = 1 + 1/2 + 1/3 + 2√(1/2 + 1/3 + 1/6) = 11/6 + 2 = 23/6
Therefore, the radius of the inner tangent circle will be 1/T = 6/23
If we apply the negative value of the root:
T = 11/6 – 2 = –1/6
Which corresponds to the radius of the outer tangent circle, which as we have seen measures 6 units, considering its curvature negative (in the case of the inner tangent circle the four circles “kiss” with their convex parts, while the outer tangent circle kisses the another three with their concave part).
From the sacred triangle to the diabolical pentacle
The golden triangle of the Egyptians is not the only sacred, golden, mystical… or cursed polygon. Without leaving the scope of triangles, the equilateral triangle would be the sacred triangle par excellence, since it represents God himself (the Holy Trinity), and not only for Christianity: for Hinduism it is also the emblem of the divine triad: Brahma, Vishnu and Shiva.
Among the quadrilaterals, the golden rectangle stands out, whose sides are in the -divine- proportion 1:1.618 (do you remember why or can you deduce it?). The familiar DIN A4 folio, measuring 210 x 297 mm, is not quite gold, but it has an interesting geometric property that makes it very special (which one is it?). Another ubiquitous rectangle is the domino or tatami, with sides in the proportion 1:2, which we frequently find in bricks and tiles, due to the advantage of being able to couple two smaller sides with a larger one to build walls or tile floors.
In the case of the regular pentagon, it is its diagonals that determine the sacred figure: the pentacle, pentagram or pentalpha, the five-pointed star venerated by the Pythagoreans. Among other geometric subtleties, the aforementioned divine proportion nests in the pentagonal star. (Can you find it?).
If we draw the diagonals of the inner pentagon of the pentacle, we obtain another one, but inverted, an inversion that is not only spatial, since the pentacle upside down is a diabolical symbol, whose curse will reach you if you do not determine the proportion between the area of the pentacle and the of its internal antipentacle. Or if you don't draw them properly, for which you have to start by drawing a regular pentagon with a ruler (not graduated) and a compass.
In the case of the regular hexagon, the construction is very simple, since the side of the hexagon is equal to the radius of the circumscribed circle, and it is just as easy to construct an equilateral triangle or a square; but in the case of the pentagon it will cost you a little more. If you succeed, try other regular polygons: octagon, enneagon, decagon… until you reach the elusive heptadecagon (17-sided polygon). Cheer up, Gauss got it when he was only 19 years old.
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#Sacred #polygons #cursed #stars