Saturday, April 14, 2018

Of Course Geometry is Magic!

I am often told that daisy wheels used in Practical Geometry are magic.  Here is my response

Yes, geometry is magic.

The technical word is 'apotropaic': these shapes are protective symbols.

The basic shapes of geometry are perfect. They never change.
So is it any wonder that we think these perfect shapes that we humans can not just imagine but also draw, are paths to the supernatural? Of course we see them as holy, sacred, mystical.

A circle of any size always comes back upon itself. Its radius, diameter, and circumference are always in the same ratio to each other. If they aren't  - it's something else, NOT a circle.
 Pi (the ratio between the circle's circumference to its diameter) is real and easy to see. Its arithmetic equivalent is infinite; it has been computed to more than 1 million digits with no end yet. (Google the number, just for fun!)

A triangle with sides 3, 4 and 5 units in length always has a 90*, square corner - seen here on the lower right side.

The square is always made up of 4 sides of equal length with 90' right angles. If any of those definitions is not present, it's not a square.

These shapes are part of each other: Here I've drawn a circle, some of its triangles, the squares that come from it.  These are the simplest forms, combined they can become endlessly varied and complex.

Geometry is science. On the grand scale geometry is the double helix of our DNA, the rotation of the planets.
It is a basic in our natural world, the small scale: the bee's honey comb, the crystals in a geode, the reflection in a mirror, the ripples of a pebble in a pond.
The Golden Section seen in the sun flower and conch shell is the expansion of these basic forms.

Too many of us found only relentless logic in our high school geometry class. We didn't twirl compasses, make daisy wheels, stars, hexagons, pentagons, octagons... using just arcs, points, and lines.
We rarely delighted in learning the magic of this world of patterns, proportions, rhythms that we are part of, that does not need words or numbers. Ionic volutes, daVinci's man, the tile in the Alhambra didn't grace our walls.

Practical Geometry is the use of geometry for construction: the arch of a Roman aqueduct or the cantilever of a suspension bridge, the vaulted ceiling and the rose window of a cathedral, the timber frame of a barn, the placement and size of architectural elements. It was used to build the pyramids, noted in the Bible. It is ancient, now mostly forgotten due to the Industrial Revolution.

I study and blog about this practical geometry, part of our heritage which we no longer perceive, to help us recover it.
I hope we will learn again to see it and use it.

For more on this c. 1830 house see:

For the use of geometry in the past several thousand years see:

Monday, April 9, 2018

Practical Geometry at MIT

 This is the curtain wall of the Mass Ave entrance at MIT,  the Massachusetts Institute of Technology, Cambridge, Mass.
The picture arrived in my mail box last week.

Immediately I saw the geometry. I knew what geometry the designer used and how it was manipulated.

I am writing three separate posts for this blog, one for my local blog. None has quite come together. Each has parts which require more drawing, thinking, and better choice of words.
Then the latest MIT mailing to alumni/ae showed up with this picture.
I laughed. I walked beneath that wall of glass and columns for 4 years. In that time I paid a lot of attention to how we used the space it sheltered, how the shape and size of that 'entrance' directed what we did. The curtain wall was not part of my thinking, although the light it allowed into the rotunda was.
The wall's pattern, rhythm, proportions - or even just the idea that it was geometry - was not part of my analysis, nor was it ever alluded to by others. 

Here is the pattern.
Upper left : A square and its diagonals.
Upper middle: The circle that comes from using the diagonal of the square as the circle's diameter.
Upper right: The square that fits around the circle.

How the pattern grows:
Top row: Overlap a circle of the same size, so that the perimeter of each circle touches the square inside the other.
Second row: This pattern can grow sideways as well as up and down.
4 squares on the right: Once established leave out the circle, continue the squares, add the diagonal, horizontal, and vertical lines.

The pattern could have started with the circle and the 2 squares fitted around it.
Using the circle as the unit  - the 'module' or 'diameter' in classical terms - is the traditional way to begin a design. (See Palladio through Asher Benjamin.)

From a photograph I cannot judge the diameter of the column.  Does it taper? have entasis? The pilaster in the corner on the right appears to be the same width as the unit I chose: the original square.
If instead the column is the unit, the module,  the circles of the curtain wall might be 3/4 or 2/3 of the module.

I have been told that the main buildings at MIT - dedicated in 1916, designed by William Wells Bosworth -  were designed using geometry. The drawings of those buildings would be well worth studying.

Practical Geometry has become an integral part of how I see buildings.  I was surprised to find that it has become a design tool for me as it was for those who used it for construction.

Meanwhile, this little bit of geometry was just plain fun.