Architectural drawings and state secrets

If you frequent the object groups of the National Museum of American History online, you may have noticed a recent addition (group link at end of post). These objects look like a cross between doll-house furniture and some strange science experiment. To explain what these intriguing mathematical objects are would be too, well, mathematical. Let me tell you a story instead.

A piece of tan-colored paper folded in half. One half is lying flat and the other is open facing the camera, like an open mouth. There are geometric drawings on the paper with strings attached. The top says "Relief 19" with text below it in French.

A young French draftsman brooded over his drawings. He was working on something new, a new way of thinking about drawing. He had come to the notice of the French authorities a few years earlier for his masterful drawings of his hometown of Beaune, in the east of France. He was appointed a draftsman at the École Royale du Génie in 1765, at age 19. But young Gaspard Monge had already shown his talents, completing his college courses by age 17. The year after his appointment, he was asked to draw plans that would prevent attackers from seeing into or firing on fortifications, while at the same time allowing defenders to have clear visibility and avoiding dead zones, locations below the fortification walls shielded from defender's fire.

Constructing ever more elaborate fortifications was the arms race of most of the 20th century. By this time in European history, star forts were the pinnacle of fortification technology, and were seen across Europe and in the American colonies. First constructed in Italy in the 15th century, star forts only became obsolete in the 18th century with the advent of high-power artillery that could breach even thick masonry walls. Castillo de San Marco in St. Augustine, Florida, is a simple yet impressive example. The complex Bourtange fortification in Groningen in the Netherlands is a stunning piece of engineering. The sharp corners that look out as well as back at the fort allow defenders complete line-of-sight to their attackers. But how to draw the plans for these increasingly complex constructions, with their walls that nowhere meet at right angles, and their tunnels and corridors inside? That is what kept Monge at his desk late into the night.

Traditional building plans consisted of what are known as plan views. A drawing would be made of what the building was to look like from each side, the top, and the floor (or its footprint). For interior spaces, the same types of drawings were made. But this left engineers and builders with no clear idea of how walls should join or stairwells connect to different levels. In particular, the geometry of how various architectural elements came together was missing. This was the puzzle young Monge stewed over at age 20. Then he hit on it. If two views could somehow be produced on the same piece of paper, maybe the angles of intersection of walls and tunnels could be shown. Monge realized that by projecting the design element both vertically and horizontally, all the geometry could be captured. Gaspard Monge had created Descriptive Geometry. It was the first time the geometry of three-dimensional objects could be captured accurately—the computer-aided design of its day.

But before it was barely out of the box, Descriptive Geometry disappeared for several years, having been quickly classified as a state secret by the French government and only taught at the French military academies. But by 1799 it was no longer classified and the first text book on Descriptive Geometry was published, Monge's Géométrie descriptive: Leçons données aux écoles normales. Descriptive Geometry quickly became an important component of the training of engineers and military officers throughout the western world, being taught at the United States Military Academy at West Point, New York, since its founding in 1802 until the advent of computer-aided drawing systems in the mid-20th century. Monge's career soared. He became the director of France's premiere engineering school, the École Polytechnique, and accompanied Napoleon on his infamous expedition to Egypt.

After Monge, several other mathematicians took up the mantle, pushing Descriptive Geometry further and writing more text books. One such author was A. Jullien, an instructor at the Lycée Sainte-Barbe, who produced the text, Cours élémentaire de géométrie descriptive. Lacking a way to effectively allow his students to visualize how three-dimensional objects are rendered in Descriptive Geometry, Jullien constructed a set of 30 handheld models, called reliefs. These delicate objects, made of wire and paper, joined the collections of the National Museum of American History in 1986. They were to be used in the classroom, following the text, just as a modern electronic text has animations or interactive applications at key points. The third edition of Jullien's text appeared in 1881 and is available on Google Books. Readers are encouraged to access the text and the images of each relief found in the object group for these models.

A flat wood box that is open that contains all of the paper diagrams that are mentioned in other portions of the post. They are lined up in four rows.

These 30 models progress from simple, showing how to render a point or a straight line…

A piece of tan-colored paper folded in half. One half is lying flat and the other is open facing the camera, like an open mouth. There are geometric drawings on the paper with strings attached. The top says "Relief e" with text below it in French.

…to the complex, the construction of a slanted pyramid.

A piece of tan-colored paper folded in half. One half is lying flat and the other is open facing the camera, like an open mouth. There are geometric drawings on the paper with strings attached. The top says "Relief 30" with text below it in French.

By looking at the image above, you can almost see, or imagine, the pyramid, represented by the strings, being projected backward onto the vertical plane, as well as downward onto the horizontal plane. When that piece of papers is laid flat, the geometry of the object, shown by the additional lines and curves on the card, can be interpreted using traditional Euclidean methods.

To better allow you to see this, I have created my own version of relief 6, which shows the construction of a line parallel to a given line through a point. The given line is the red string on the right and the point through which the parallel is to be constructed is shown as the bend in the wire on the left at point (m, m'). The constructed parallel is the red string on the left.

A piece of tan-colored paper folded in half. One half is lying flat and the other is open facing the camera, like an open mouth. There are geometric drawings on the paper with strings attached. The top says "Relief e" with text below it in French.

In the image below, I have built my own version of relief 6 out of the (apropos) empty box my migraine medicine comes in.

A white cardboard box with one of the side cut away. There are geometric drawings in the inside of the box with string and small letters to label the angles.

Below is the paper the projections were drawn on laid flat, as it would appear in an architectural drawing or Descriptive Geometry textbook. In this form, the angles at which the lines meet the vertical and the horizontal planes can be seen, giving the designer or architect clear details of the geometry of the object being portrayed. The horizontal line at Q is the fold, and the strings would run from f' to e and d' to c.

A photopgraph of a white piece of paper that resembles the models seen throughout the rest of the post. There are lines drawn in pencil intersecting each other at various points with small letters next to angles.

In current mathematical, engineering, or design courses, three-dimensional objects are rendered using any one of several lightning-fast computer algebra or drawing systems such as MATLAB or Mathematica. But teachers have always sought ways to bring mathematical objects to life for their students. Wood, metal, and plaster models have been used in the classroom for centuries. The museum has approximately 30 mathematics-related object groups, several of which are of educational models for you to explore. But rarely are classroom models of such ingenuity and delicacy as the Jullien Models of Descriptive Geometry.

They are so beautifully made they reminded me of doll-house furniture the first time I saw them.

To learn more and explore the full set of Jullien models for Descriptive Geometry, visit our online object group.

Dr. Amy Shell-Gellasch, who volunteers at the museum, is an historian of mathematics and Associate Professor of Mathematics at Montgomery College in Rockville, Maryland.