Roof Framing Geometry

Euclid's Elements Book XIII Proposition 17

I have always been fascinated by “who knew what and when they knew it. “ As well as how they knew it. So, yesterday I was searching for a download link on the book De quinque corporibus regularibus by Pierodella Francesca. That has the Platonic Solids and the Archimedean solids. I had already downloaded the book Divina proportione by Luca Pacioli. That supposedly plagiarized Piero della Francesca book. I saw in Luca Pacioli book that he had drawn Euclid's Elements Book XIII Proposition 17, but not Euclid's Elements Book XIII Proposition 18. So, I was wondering if Piero della Francesca had a drawing of Euclid's Elements Book XIII Proposition 18. I never did find a link to download Piero della Francesca's book. Even though I read on a page at the JLC forum, I thought I should study Stereotomy by reading the book De quinque corporibus regularibus by Piero della Francesca 15 years ago.

Over the last two months, I’ve been studying Euclid. Stereotomy is taking a point in 3D space and laying down that 3D point on a 2D drawing. So you can scribe the wood. After studying Euclid for the last two months, I realized I needed to take the next step in my journey in geometry and mathematics. Where you take a point in 3D space and transfer that point to another point in 3D space.

Hippasus, who was part of the Pythagorean Brotherhood, a secret society of mathematicians, could draw the Platonic Solids, the Icosahedron or the Dodecahedron. I don't know which one it was. But he was stoned to death for having too much knowledge in 500 BC.

My goal in my journey in geometry and mathematics is to learn Critical Thinking in Geometry so I can lay down the 3D points in space like Hippasus and Euclid. Euclid's Elements Book XIII Proposition 17 is an excellent example of Critical Thinking in Geometry.

Representing a three-dimensional 3D point in space, the objects that compose it are studied analytically, in their shapes and position in relationship to the place that contains them.

My study on nesting the Platonic Solids is based on Euclid's proposition 18. Here I'm able to draw out the edge lengths of the 5 Platonic Solids in a sphere using 1.618 and √2.

Tetrahedron, Cube, Octahedron, Icosahedron, and Dodecahedron

Sphere Radius = EC = AC = 1

FB side of Cube = 1.15470

EB side of Octahedron = 1.41421356

MB side of Icosahedron = 1.051462

AF side of Tetrahedron = 1.632993

QB = NB side of Dodecahedron = 0.713644

(√5 + 1)÷2= 1.618033

2÷√3 = 1.1570

√2 = 1.41421

√(2- ((2√5 )÷5))= 1.051462

2÷√3 *√2 = 1.632993

(2÷√3 ) ÷ ((√5 + 1)÷2)= 0.713644

Euclid's Elements Book II Proposition 11

Extreme and Mean Ratio

The Golden Ratio

To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

Triangle ABE is the √5 triangle, and triangle GFC is the Dodecahedron Hip Rafter Slope.

Euclid's Elements Book XIII Proposition 18

To set out the sides of the five figures and compare them with one another.

Set out AB the diameter of the given sphere, and cut it at C so that AC equals CB, and at D so that AD is

double DB. Describe the semicircle AEB on AB, draw CE and DF from C and D at right angles to AB,

and join AF, FB, and EB. Then, since AD is double DB, therefore AB is triple BD. In conversion, therefore, BA is one and a half times AD. But BA is to AD as the square on BA is to the square on AF, for

the triangle AFB is equiangular with the triangle AFD. Therefore the square on BA is one and a half times the square on AF. But the square on the diameter of the sphere is also one and a half times the square on the side of the pyramid. And AB is the diameter of the sphere, therefore AF equals the side of the pyramid. cube.

Euclid's Elements Book XIII Proposition 17

To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to

prove that the square on the side of the dodecahedron is the irrational straight line called

apotome.

BCEF is the top of the Cube. H,N,M,O are midpoints of the square on top of the cube. Draw lines HM, NO, GK, and HL. Mark off point J, the midpoint of HP, and draw line OJ. Then draw line JY the length of OJ. Develop the square PYIS. Point S will divide the line PO, apotome, into the Golden Ratio. PS is the extreme, and SO is the mean. Mark off points UXV perpendicular to the plane of the cube, using the length PS. Mark off point T, using the length SO. Mark off point W perpendicular to the plane on the side of the cube, using length PS.