A lesson in Applied Geometry and Euclidean Geometry

Euclid of Alexandria (200 BC) Greek 
Archimedes of Syracuse (287 – 212 BC) Greek
Apollonius of Perga (262 – 190 BC) Greek 
Marcus Vitruvius Pollio (80 – 70 BC) Roman
Anthemius of Tralles (474 – 558) Greek - Roman
Abul Wafa al-Buzjani (940 – 998) Persian, Baghdad 
Villard de Honnecourt (1225) Cistercian Order of France
Matthäus Roritzer (1435 – 1495) German
Albrecht Dürer (1471 – 1528) German
Rodrigo Gil (1500 – 1577) Spanish
Andrea Palladio (1508 – 1580) French
Philibert De l'Orme, (1515 – 1576) French
Francois Derand (1588 – 1644) French 
Mathurin Jousse(1607 – 1692) French
Gérard Desargues(1591 – 1661) French 
Amédée-François Frézier(1682 – 1773) French
Gaspard Monge (1746 – 1818) French
Peter Nicholson (1765–1844) British 
Asher Benjamin (1773 – 1845) American 
Robert Willis (1800 – 1875) British

Last summer (2010) I became interested in medieval Gothic cross ribbed vaulting after reading and seeing “Pillars of the Earth”, by Ken Follett. I wanted to know if the Medieval thru Renaissance builders used a trammel or a rope to draw out the ellipses of the cross vaults. Or did they use ordinates to transfer the points of the circles (major/minor) to points on the ellipse? What I ended up getting was a history lessons in geometry. I ended up with more questions, than answers to my question. The following is a brief summary of the history lesson I got in geometry.

A Gothic vault is composed of ribs, keystones and webs (curved masonry that fills the web between ribs). The ribs are the cross vault diagonals, transverse ribs and the wall ribs.

There are hundreds, if not thousands of architectural historians over the last two hundred years who have basically asked the same question. Or more importantly, how did the medieval cathedrals get built. A lot of the architectural historians provide a good insight into the building of the cathedrals, but in the end their findings are speculative. Do to the lack of documentation on the medieval construction techniques. As a carpenter-builder I've read a couple hundred documents, thesis and books by the architectural historians and my findings are as speculative as theirs, but thru the eyes of a carpenter who uses a lot of the same building techniques as the medieval carpenters did 1000 years ago.

I’ll start with “the following diagram is self-evident and needs no further explanation”. I’ve read the self-evident statement in several of the “Carpentry and Building Journals” published from 1880 to 1905. It also appears in most of the books written on carpentry from 1800 to 1900. What the writer’s were really saying was that they were expressing postulates just like Euclid that were self-evident. There are really only two postulates from the Euclid's Elements that we need to know as carpenters to end all of our paragraphs with “it is self-evident and needs no further explanation”. The two postulates are Euclid’s Elements Book 1 Prop 1 and Euclid's Elements Book 1 Prop 47.

Euclid's Elements Book 1 Prop 1, to construct an equilateral triangle on a given finite straight line A-B. The radius of the circles are AB. The equilateral triangle ACB is formed by the intersection of the two circles.

This is the basic geometric figure that 100’s of other geometric figures can be drawn from. Two easy examples that we use as carpenters are the equilateral arch and the Gothic Arch.

Equilateral Arch from Euclid’s Elements Book 1 Prop 1.

Gothic Arch from Euclid's Elements Book 1 Prop 1.

Euclid's Elements Book 1 Prop 47. In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. We've all been there done that with the Pythagoras theorem, but it's neat to count the squares so we can say "it is self-evident and needs no further explanation".

From Euclid's Elements Book 1 Prop 1, to construct an equilateral triangle you can easily create 100’s six point geometric figures.

To draw the Euclidean six point geometry start with radius line AB on two perpendicular lines. Set your compass at B and draw the circle with radius AB. Next set your compass at point 1 and using the same radius AB swing the compass to draw the first arc that touches the sides of the circle. Next set your compass at point #2 where the first arc touches the circle and swing another arc. Procede around the circle setting your compass at the intersection of the arc and circle to draw the rest of the arcs.

The six point geometry can also be call six rose petals, seed of life, timber framing daisy wheel, Dutch Hexagram, thunder marks, hex rose petal or the basis for the Star of David or that King Solomon’s seal is based on six point geometry. The Vesica Piscis is also derived from Euclid's Elements Book 1 Prop 1. The six petal flower design may also be a signature, left behind during the passage of an apprentice geometer or apprentice mason, one who knows the road to Euclid. When you see the Euclidean six point geometric drawing on the Knights Templar tombstone at St Magnus cathedral in Kirkwall, is it a six petal flower design, daisy wheel or a religious hex symbol, or the seed of life symbol or just six point geometric construction?

Vesica Piscis developed from Euclid's Elements Book 1 Prop 1.

Ad Triangulum rotated equilateral triangles from six point geometry

If I get a chance to talk to French carpenters, who belongs to the French compagnons carpenter guild system, I can say yes, I’ve read-looked at one of the first book(in French) on carpentry by Philibert De l'Orme, (1515 or 1576) LE TROISIEME LIVRE DE L’ARCHITECTURE . Philibert De l'Orme discusses the geometry of Euclid, Archimedes and Vitruvius. By studying De Architectura by Vitruvius, known today as “The Ten Books on Architecture”, a treatise written in Latin and Greek on architecture, Philibert De l'Orme uses symmetry and proportion that was based on Ad Triangulum (Equilateral, Hexagonal base) and Ad Quadratum (Rotated Squares, Octagonal based). Philibert De l'Orme also discusses masonic projection (stereotomy) and his new invention, vault stick framing. Up until 1515, most roofs were timber framed.

Ad Triangulum (triangle within the circle)
Ad Triangulum (hexagonal base)
Ad Triangulum (√3 base)
Ad Triangulum (three point geometry)
Ad Triangulum (six point geometry)

Ad Quadratum (square within the circle)
Ad Quadratum (octagonal based)
Ad Quadratum (√2 based)
Ad Quadratum (four point geometry)
Ad Quadratum (eight point geometry)

Ad Quadratum rotated squares from 4 point geometry

Philibert De l'Orme layouts the 8 wind directions for the orientation of a building based on Ad Quadratum.

Philibert De l'Orme layouts the spherical cross vault based on Ad Quadratum.

Philibert De l'Orme was the inventor of the first speed square based on the Equilateral Triangle of Ad Triangulum.

Philibert De l'Orme explains how to plot the points of a lengthened arch (ellipse) from the 3 lost points of the arch. Points A & B are the mid points of the chords and the point C is the center of the circle located by the perpendiculars of the chords. By suggesting that the 3 points were lost, Philibert De l'Orme is saying that the builders he came in contact with had forgotten their geometry. By calling the arch lengthened instead of an ellipse, it shows that in the early 16th century (1515) no one had a clear understanding of the mathematical properties of an ellipse. Leonardo da Vinci, Albrecht Dürer, Hernán Ruiz and Mathurin Jousse also draw the ellipse with ordinates but call it a lengthened arch [SANTIAGO HUERTA 2009]. However, Francois Derand in 1643 clearly shows a trammel for an ellipse in his drawings. Another interesting description, by Philibert De l'Orme, for developing the lengthened arch(Cherche rallongée) is the use of "dropping perpendiculars" or "crawling lines". So I don't think we'll find the use of the word "ordinates" in medieval construction, even thou they developed their drawings using what we call ordinates. Also, Philibert De l'Orme does not call it a new invention. So the medieval builders must have known how to drop perpendiculars and crawling lines to develop the lengthened arch.

Daisy wheel used by medieval timber framing carpenters.

Vitruvius Roman Theater based Rotated Equilateral Triangles Ad Triangulum Geometry

Vitruvius Greek Theater based on Rotated Squares Ad Quadratum Geometry

Vitruvian Man -- Little Green Men with Correct Proportions

Symmetry arises from proportion according to Vitruvian principles (70 BC). So when you see a picture of the Vitruvian man by Leonardo Da Vinci (1451), you know he based his drawing on Vitruvius, Book III, Chap. 1, on the circle and square of Ad Quadratum. A square (homo ad quadratum), that is placed in a circle (homo ad circulum). "Without symmetry and proportion there can be no principles in the design of any temple; that is, if there is no precise relation between its members, as in the case of those of a well shaped man." Should the elevation of our houses have the proportions of a well shaped man or should they be based on the golden ratio? Or Ad Triangulum Geometry? Can we say that our house is based on Ad Quadratum Geometry if our house is square or has octagonal geometry?

Golden Ratio Framing

Empirical geometry, information gained by means of observation, experience, or experiment, used by masons and carpenters over the last 2000 years is the root of applied geometry, namely the branch of geometry that we call descriptive geometry today. Give a medieval mason or carpenter a compass and the techniques that they used to draw out curved complex cuts are called stereotomy. Descriptive Geometry is the theoretical side of stereotomy. Is stereotomy based on the eleventh and twelfth books of the Elements of Euclid? Or was stereotomy (masonic projection) just a building technique by the masons and carpenters ?

The method of Ad Quadratum that is described by Roriczer in 1451 where he takes a square and draws a diagonal square inside and in a series of further steps gives the technique for "elevating" a cathedral pinnacle. This general Ad Quadratum technique was used to produce other structures including the cross-sections of buildings. Another medieval technique of taking the floor plan to the elevation is based on the equilateral triangle (Ad Triangulum). The first modern exercise in descriptive geometry wasn't done until 1471 by German painter Albrecht Dürer. Albrecht Dürer called the descriptive geometry of the ellipse from a conic section, oval, or the egg line shape. So did the medieval carpenters draw ovals instead of plotting out the ellipse with ordinates? There have also been Egyptian cave drawings of the geometry to draw out ovals.

The difference between an ellipse and oval is so small that no one can really tell the difference between an ellipse and oval without measuring the oval.

Before the 16th century, builders only had extremely simple construction geometry techniques. Villard de Honnecourt’s sketches in 1250 showed crude geometric forms of Euclidean geometry and were-are supposedly the basis of freemason’s secret [Marie-Thérèse Zenner 2003]. It wasn’t until Gaspard Monge, wrote about "Descriptive Geometry, or the Art & Science of Masonic Symbolism”, in 1776 the secret word was completely revealed.While the Romanesque builders liked the symbolism of numbers that referred directly to the biblical message and its interpretation, medieval masons and carpenters revered above all, the geometry of Euclid from compass and straight edge.

Francois Derand (1588 -1644) published LES LIVRES D’ARCHITECTURE, after Philibert De l'Orme, (1515 or 1576) LE TROISIEME LIVRE DE L’ARCHITECTURE . Francois Derand, “The architecture of arches” could be directly related to the descriptive geometry we use today to draw out roof framing kernels, ellipses or groin vaults using ordinates.The secrets of the professionnel builders, des maçons. Andrea Palladio, a French architect also stuied Vitruvius and used the ellipse in his designs. Some of Andrea Palladio drawings are egg shaped, while others are true ellipses. It's hard to tell if Andrea Palladio had any real influnce over the construction techniques of the Medieval - Renaissance builders. Two other French architects/ mathematicians who might have influnced the construction techniques of the Medieval - Renaissance builders are Gérard Desargues and Amédée-François Frézier. Any of these French architects/mathematicians could have been the basis of the “British System of Projection”, by Peter Nicholson in 1832 who could be considered the father of American roof framing geometry. However, the French compagnon carpenter guild system might argue that the roof framing geometry we use today is based off of the book by Mathurin Jousse (1607–1692 French) or other books that the French aren’t sharing. Louis-Mazerolle's (1889) Traite Theorique Et Pratique De Charpente and Art Du Trait Pratique De Charpente by Emile Delataille (1890) are good examples of stereotomic development at its peak by French carpenters. Without being able to read French or German, I’m only speculating that the stereotomic drawings of Francois Derand are more precise than Philibert De l'Orme drawings, thus easier to develop other construction geometric drawings from that were the basis of our roof framing geometry. [updated] "Traict Five Orders of Architecture" translated from Palladio by Pierre Dumb in 1645 has a lot better geometric roof framing drawings. It also has a Gambrel Roof drawing. The Dutch published a counterfiet copy of the book in 1646, it looks like the Dutch Gambrel Roof is a Counterfeit French Mansart Roof in more ways than one.

Stereotomic projection By Frezier

Orthographic projection & Stereotomic projection

Marcus Vitruvius Pollio, in De Architectura (80 BC), Book I, writes about orthography and scenography, (orthographic projection and stereotomy projection). Stereotomy relies on the use of horizontal and vertical projections in determining in two dimensions the precise configuration of the 3 dimensional complex parts of a building.

In the 10th century Persian mathematician Abul Wafa al-Buzjani at Baghdad wrote the treatise "On Those Parts of Geometry Needed by Craftsmen". Abul Wafa al-Buzjani writes about six point geometry and other types of geometry needed by craftsmen. You can see some of this geometry each time you step on your Persian rug and count the 3, 4, 5, 6, 7, 8, 10, 12, 16 or 20 point geometric shapes in the rug. Islamic geometry shows the influence that Euclid, Vitruvius, Archimedes had on Islamic mathematics after the late 8th century translation of Euclid's Elements into Arabic in Baghdad. Archimedes and Apollonius of Perga books on Conics, who named the conic section, ellipse, parabolas, hyperbolas , were also translated from Greek to Arabic and may be the reason for the medieval builders using conic and spherical geometry for their vaults instead of the Romanesque barrel vaults or groin vaults. Was medieval French stereotomy based on the orthography and scenography that Vitruvius describes, or was it developed-influenced by the books from Baghdad?

Why is the colonial crown molding spring angle at 38° - 52°. Is it based on the golden ratio of phi? (1 + √5)/2 = 1.618033989. The arctan (√phi) = 51.83°. The inverse slope of 90° - 51.83° = 38.17°

Circle inscribed in a 1:2 rectangle produces the golden ratio of phi and the 51.83° slope of the pyramid

Egyptian pyramids base (b)= 1, √phi = height(h), phi = apothem(a), slope angle = arctan(√phi ) = 51.83° . The Romans kept the golden ratio as the uniform proportion in their construction (major axis/minor axis of the ellipse), width and height of steps, (tread & riser) width and height of the rows around the Coliseum of Rome.

Formula for a golden ratio ellipse

Minor axis = X

Major axis = X * 1.618033989

Formula for a golden ratio stairs

Stair Riser = X

Stair Tread = X * 1.618033989

(7” riser = 7 * 1.618033989) = 11 5/16” tread

Formula for a golden ratio Roof slope

Rafter Run = X

Rafter Slope = X * (1.618033989 – 1)

(12” run = 12 * (1.618033989 – 1) = 7.416= 7 7/16 = 31.72°)

Or is the colonial crown molding spring angle based on the umbra recta shadow?

The Greeks and Romans developed sundials based on the length of the shadow cast by a stick-gnomon. The umbra recta simulates the shadow cast on the horizontal plane by a vertical gnomon when the Sun's ray is inclined between 0° and 45°. The umbra versa simulates the shadow cast on the vertical plane by a horizontal gnomon when the Sun's ray is inclined between 45° and 90°. When the Sun’s ray is inclined by 45°, the two shadows are equal (umbra media). 

Vitruvius - Ptolemy Alalemma-Sundial from 12 point geometry and orthographic projection

Vitruvius the Alalemma Sundial and Its Applications Book 8, Chapter 7 The ellipse, and its centre, is the point at which all Elliptical Geometry begins. By plotting abscissas and ordinates of the sun's shadow, you can drive the chariot of the sun's arc south of the equator to see the chariot of light at winter solstice. After developing the drawing from Vitruvius “The Alalemma Sundial and Its Applications” Book 8, Chapter 7, I’m amazed at the geometry that was developed over 2000 years ago. The Elliptic “Alalemma Sundial” would have been developed using ordinates. So the use of ordinates would have been general knowledge to the medieval builders to build an elliptic sundial who wanted to know when to stop work for the day to head to the ale taverns to have strong beer after drinking weak beer for breakfast and lunch along with their stale horse bread. Conclusion It could take years to read and analize all of the books written on vaulting. Robert Willis wrote the book " On the Construction of Vaults of the Middle Ages" where he discusses Gothic vaulting techniques and the stereotomy by Philibert De l'Orme, Francois Derand, De la Rue, Mathurin Jousse, Amédée-François Frézier and others. Willis said the Gothic rib vaulting used only circluar arcs and no elliptical vault ribs were used. An easy way for the medieval builders not to know anything about drawing out an ellipse. Philibert De l'Orme and Abul Wafa al-Buzjani both criticized the masons and carpenters for sloppy construction due to a poor understanding of geometry. Probably only 10% of the medieval masons and carpenters really knew their geometry, which is no different than today. I’m pretty sure if we ask the carpenters of today to plot an ellipse using the abscissas(X) and ordinate(Y) of an ellipse, you’ll get a blank stare of fright. Even thou it’s called “the trammel of Archimedes”, there’s no documented evidence of it being used before the 16th century and it would have been tough to build a trammel for a 32’ or 48’ major axis. The use of a rope or string for drawing an ellipse was first brought to light in the 6th century by Anthemius of Tralles , a Greek professor of Geometry in Constantinople, where they built 336 groin vaults underground in Constantinople for water storage  However, there is never any real documentation of the medieval builders using the string method. With a lot of the medieval vaults based on conic and spherical geometry the rope or string or trammel method would have been useless on parabolas, hyperbolas or double curves surfaces from the intersection of a cone and sphere. More questions arise than answers, but at least I now know, according to Vitruvius, that towers should be either round or polygonal, never square. And that the distance between each tower should not exceed an arrow's flight. A cubit is a forearm, from the elbow to the tip of the fingers.

Notes: I first wrote this article for an American carpenters magazine and I didn't include foot notes. I was then told it was too intellectual for American trade magazines. As I update this article I'll add the foot notes in [brackets] for the architectural historians.