Saturday, February 2, 2013

Regulating lines of Geometry Without the Assistance of Real Numbers

I've been  following Jane Griswold Radocchia ,Architect, blog on regulating lines of geometry . She asked me to expand on my comment on her blog Carpenter squares in1503.

My comment:


What's fun about these drawings are the platonic solids on the table in the drawing. John Brown, 1661, also had the platonic solids in his portrait/drawing with his carpenter's rule/square.

A Note on the Wooden Carpenter’s
Rule from Odyssey Shipwreck Site 35F


Expanding on my comment:

There are exactly five regular polyhedra (platonic solids).
Euclid's book of Elements book XIII

The 5 Platonic solids: 
The Tetrahedron (3 equilateral triangles at each vertex) 
The Hexahedron (3 squares at each vertex, cube) 
The Octahedron (4 equilateral triangles at each vertex) 
The Dodecahedron (3 pentagons at each vertex) 
The Icosahedron (5 equilateral triangles at each vertex) 

Compound Saw Miter and Saw Blade Bevel angles for building the 5 Platonic Solids out of wood Wood polyhedron or Wooden Polyhedra

John Brown an mathematical instrument maker, 1661, had the platonic solids in his portrait/drawing with his carpenter's rule/square.

John Brown portrait with the platonic solids.

Pepys worked with John Brown to design a custom-made rule, which Pepys called the “most useful that ever was made, and myself had the honor of being as it were the inventor of this form of it.”

The Diary of Samuel Pepys

The mathematical practitioners of the seventeenth century all subscribed to the Vitruvian   vision of  the architect as a geometrical paragon. With the platonic solids in the drawings/ portraits it showed how the  mathematical practitioners studied Euclid's books of  Elements.  The mathematical practitioners were in the process of developing Analytic geometry, ( Descartes. In Book III of La Géométrie ) but wanted to show how their mathematical instruments were based on solid geometry already proven by Euclid's propositions.

Elucidation of the stuff is self evident. 
We hold these truths to be self-evident , and need no further explanation. 
Stuff = roof framing geometry
Stuff = tangent handrailing geometry
Stuff = Regulating lines of Geometry
Stuff = Ad Quadratum , square rotated in circle
Stuff = Ad Triangulum , 
Stuff = Daisy Wheel

Architecture and Mathematical Practice in England, 1550-1650

The mathematical practitioners wanted to introduce to the English skills that were monopolized by foreigners.  This must be in reference to the  Islamic Moors . Such as the Alhambra, Granada Spain.

The roof over the Hall of the Abencerrajes is unique because the roof actually follows the Ad Quadratum ground plan that is used to develop the octagonal ground plan. Squares rotated in the circle.

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