7/27/2019 Construct Paperpyramid(Cheops)

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Geometric Construction of the Golden RatioSubdivide a square of side 1 into two equal rectangles. Then lay out a distance equal to the diagonal of one of these half-squares, plus half the side of the original square.

The ratio of this new distance to the original side, 1, is the golden ratio.

Project: Do this construction for the golden ratio.

Project: Mathematically show that this construction gives the golden ratio.

Egyptian TriangleLet's now return to the pyramids. If we take a cross-section through a pyramid we get a triangle.If the pyramid is the Great Pyramid, we get the so-called Egyptian

Triangle.It is also called the Triangle of Price, and the Kepler triangle.

This triangle is special because it supposedly contains the golden ratio. In particular,

the ratio of the slant height sto half the base bis said to be the golden ratio.

To verify this we have to find the slant height.

Computation of Slant Height sThe dimensions, to the nearest tenth of a meter, of the Great Pyramid of Cheops, determined by various expeditions.

height = 146.515 m, and base = 230.363 m

Half the base is

230.363 2 = 115.182 m

So,

s 2= 146.515 + 115.182 2= 34,733 m2

s = 18636.9 mm

Does the Great Pyramid contain the Golden Ratio?Dividing slant height s by half base gives

186.369 115.182 = 1.61804

which differs from !(1.61803) by only one unit in the fifth decimal place.

The Egyptian triangle thus has a base of 1 and a hypotenuse equal to !. Its height h, by the Pythagorean theorem, is given by

h2= ! 2- 12

Solving for hwe get a value of !.

Project:Compute the value for the height of the Egyptian triangle to verify that it is !.

Thus the sides of the Egyptian triangle are in the ratio

Kepler TriangleThe astronomer Johannes Kepler (1571-1630) was very interested in the golden ratio. He wrote, "Geometry has two great treasures: one is the theorem of Pythagoras,

the other the division of a line into mean and extreme ratios, that is , the Golden Mean. The first way may be compared to a measure of gold, the second to a precious

jewel."

In a letter to a former professor he states the theorem, which I rephrase as:

If the sides of a right triangle are in geometric ratio, then the sides are

We recognize this as the sides of the Egyptian triangle, which is why its also called the Kepler triangle.

Project:Prove that If the sides of a right triangle are in geometric ratio, then the sides are

A British railway engineer, Robert Ballard, saw the pyramids on his way to Australia to become chief engineer of the Australian railways. He watched from a moving

train how the relative appearance of the three pyramids on the Giza plateau changed. He concluded that they were used as sighting devices, and wrote a book with the

grand title of The Solution of the Pyramid Problemin 1882.

He also noted that the cross-section of the Great Pyramid is two of what we have called Egyptian triangles. He then constructs what he called a Star Cheops, which, he

says, "... is the geometric emblem of extreme and mean ratio and the symbol of the Egyptian Pyramid Cheops."

To draw a star Cheops:

Draw vertical and horizontal axes.

Using their intersection as center, draw two circles, radius 1, and radius 1 +!.

Superscribe a square about the smaller circle. This will be the base of the pyramid,

From the point where an axis cuts the outer circle, draw two lines to the corners of the square.

The triangle obtained will be one face of the pyramid.

Repeat the preceding step for the remaining three faces, getting a four-pointed star. Cut it out.

Fold each triangular face up from the base forming the pyramid.

ProjectDraw a star Cheops. Fold it to quickly make a model pyramid.