Wheels of Fire: 12th Grade Math in Architecture

CAKE MOLD OR CALVARY

Maths in Architecture – Class 12 – Middle Lesson

Did the idea of providing a main lesson on Architecture to Class 12 occur to Rudolf Steiner as he struggled with the labyrinthine complexities of his First Goethean? Or years later when considering the needs of the students in his new high school? Whatever, this exciting, enlightening 3-week unit has become, over the last 8 decades, one of the cultural climaxes of his remarkable education.

The programming is not obvious: Could Architecture be a social science? A science even? Art history? I have included it in the Maths stream; not that this First Art (as the word means) does not appear often and effectively in all the above, but because all structural-engineering elements in a building enshrine number principles, often of a sublime dimension. For more cultural content than this “Maths in Architecture” chapter can provide, see my book Many Mansions.

In the Educational Zodiac, starting with Cancer in Class 1, Class 12 have reached the pinnacle from which they can confidently launch themselves into adulthood. Gemini is the inspiring sign for 18-year-olds, with its Sense of Ego and Quality of Faculty. Both these are especially relevant in this subject; the Ego lives in the Physical Body, as the Ego-borne mind is manifest in a building’s structure. The opposite from the sense, which impresses from without in, is the quality, an expression from within out. Faculty is of course called upon in every phase of building, from the first conceptual sketches to the last lick of paint. We can know much of the nature of both Rudolf Steiner’s Ego and Physical Body by studying his First Goethean (and to a degree in the second).

How on earth did he find the time, amongst all his other obligations, to design and supervise the construction of the most profound example of spiritual architecture of the 20^th Century? In my view, the Sydney Opera House is the cultural equivalent. Curiously both are dominated by shells; these are expressions of the physical (or spatial as Steiner sometimes calls it) body aspect of this Art of the Physical Body.

The shell is a skull, the human head is the physical body element of 4-fold man. The First Goethean was called “The House of Speech”; the sense of word is ascribed to Aries; so to travel full circle, the region of The Ram is the skull. The event on Calvary, meaning ‘the place of the skull’, occurred in the vernal sign of Aries.

In one jocose description, Steiner refers to the two semi-domes of his building as ‘cake molds’. This is to emphasize that the building is merely a shell, the cake is the living content which it shelters. The Opera House is a House of Song, especially as this relates to opera, representing yet a different voice in the many-cadenced choir which is the sense of word.

In the 40-fold evolution of the human being through life, 18-year-olds are enjoying their Conceptual-Pictorial Aspect of the Astral Body year – the ‘Spirit Self’ Year. The art of spirit self is Poetry, or the speech arts generally. How apt then to engage the students in such a strong ‘thinking-about-pictures’ activity as Architecture. Utzon’s first vision of the Opera House was pure concept-picture.

Due to its cultural-hear implications, Architecture is programmed as middle lesson. Of the four middle lesson streams (Professions, Cultural, Service, and Industrial), it fits most comfortably in the 4^th, the ‘physical body’ middle lesson! Of the three Industrial strands, it appears naturally as a Secondary Industry. A simple definition of secondary is making something from original (primary industry) raw materials.

All structural principles devised by man, and called architecture, can be found in human anatomy. Many designs are influenced by the extension of the human being, nature, or even from constructions of the animal world, like a wasp’s nest. In respect to human form and proportion, almost all temples from the ancient world enshrined timeless human ratios; celebrated examples being Chartres, Karnack and Ankor Wat.

And ‘timeless’ they are, both literally and metaphorically. The word temple means “to cut time”, as in time stands still. This is the archetypal experience one has in the presence of one’s sculpted deity, choir or worship ritual. Time is of life, and the happy cacophony of life is often a blinding and deafening of the spirit; that which dwells beyond the life of ether zone. A place of worship can still this etheric freneticism.

The etheric, the universal realm where all gods (meaning ‘good’) meet, is the macrocosm; the conduit to this high realm is the temple, church or sacred site, the mesocosm, and finally the body-soul-spirit human being is the microcosm; a humbling if not dis-consoling thought.

The more man detaches from nature (the Druids worshiped under the stars), the more we need to construct habitats in which to shelter from it. The structures of primitive man are much more organic than modern edifices. They often reflect a strong influence, or direct imitation even, of natural forms, such as the beehive grass huts fashioned on weaverbird nests. Forms and materials are related to the environment in ways our own building designers might profitably consider. These wise buildings also contain sound structural principles, as seen in the sturdy flat-roofed abode house of American Indians, or the steeply raked roofs of New Guinea long-houses.

12 PRINCIPLES OF STRUCTURE

This design dozen is based on just two basic powers, Tension, as expressed in the muscular system of man, and Compression, in the skeleton. These respectively expansive and contractive (Luciferic and Ahrimanic) forces are evident in all creation. They are the pull-=push dynamics found in all building; emphatically expressing in concrete (compression) and steel (tension). In the 12 Principles of Structure, there are six of each.

COMPRESSION

1. TRUSS (tie). This is a single unit of framework made from separate elements. In nature we see it in the structural integrity of tree limbs; and in human anatomy, in the cancelli of the femur, designed as it is to carry the weight of the whole upper body.

Below are two truss force diagrams. IN the first, the arrows are the resting points for purlins.

So as not to confine our minds in two mutually exclusive conceptual cells, the following example shows how, even in a truss, the two principles of force and form are structural complements. Compression, or Force, is of the will, and influences the future; tension – form – is rather of thinking, and is a mirror of the past. The forces of yesteryear created the forms of the now; the forces of the now fashion the forms of tomorrow. The inner kinship between force and form is also related to center and periphery, a specific triangle – form-periphery – is created by drawing it at right-angles to three (this works with any polygon) force-centric lines. This is a triangle of forces. The lengths of the three sides relate to the relative intensity of the forces. The triangle represents the periphery of a plane field, the Tri-Star, the force-center point.

An example of the triangle (polygon) of forces application from engineering-architecture is where three or more girders meet at a number of junctures. At each of these, the several pressures or tensions must hold each other in balance. In drawing a triangle of forces for each of these, it is not necessary to begin again with a separate figure for each point. Rather “graphic statics”, or reciprocal force diagrams, are drawn. Below is the main structure of a bridge spanning a ravine; beside this, it is reinterpreted with a graphic static diagram.

The downward arrows indicate equal weights, localized at the joints. The upward slanting arrows are the supporting thrusts at the abutments on either side. The force diagram has a line at right-angles to every line of the real structure, including also the lines of the external forces.

The length of every line in his diagram is proportional to the relative intensity of force – pressure or tension – along the corresponding line in the ‘real’ diagram. Hence, if the weights to be borne, indicated by the downward arrows, and the directions of the supporting thrusts are assumed, the magnitudes of the latter, and of the stresses, in all the girders can be directly measured; according to a freely chosen scale.

There are simple methods of ascertaining whether the stress in any given member is compression or tension – whether it is acting as a strut or as a tie. The radial and peripheral relations of the two diagrams, the one representing the visible structure, the other the relation of its unseen forces, is evident, among other things, form the lettering. In both diagrams we see the same letters – A to J; L, M, N, P, Q, R; and O. But while the letters in the ‘real’ bridge have been assigned to the plane fields – triangular or otherwise, enclosed between the girders, or between these and the lines of the external forces which they must ultimately bear, in the forces diagram they are assigned to the points.

The girder, for example, forming the boundary between the fields B and C in the ‘real’ bridge corresponds to the line joining the points B and C in the forces bridge. The relative length of this line tells us the magnitude of the thrust in this girder. From the force diagram we see which of the girders are most important, bearing the main thrusts or tensions. Those that are more auxiliary, like AB or CD, are very short lines in the force diagram. That this is so is evident to common sense when actually looking at a bridge. The method translates into scientific precision what a practical person will know instinctively.

The symmetry of the diagram is due to the symmetry of the distribution of assumed weights. But if a heavy lorry is standing on the bridge, say at the point between M and N, the thrusts at the abutments will have to change direction to accommodate the asymmetrical resultant weight, requiring a new diagram.

The outer forces (weights and supporting thrusts) bearing radially in upon the actual structure, correspond to the lines forming the periphery, the enveloping triangle in the force diagram. Astonishingly the relation of the two diagrams is in essence mutual. We have lettered plane fields in the forces picture, lettered points in the real. From the forces diagram, representing an engineering structure with certain forces acting upon it, a working model could be made. The lengths of the lines of the actual picture would then tell the relative stresses in the other.

This is way force diagrams are called “reciprocal”. What in one is invisible force, the other translates into outward form; but if the latter were the real thing, subject to forces which it holds in balance, the former by its visible proportions would represent the unseen stresses in this one. The relationship of an engineering structure to its force diagram is closely related to the point-plane polarity of Projective Geometry.

2. COLUMN. This can be any supporting pillar, seen in human anatomy in the femur, which hold up the weight (compression) of the whole upper body. An obvious column form nature is a tree truck.

The concept of a column is based on two principles, movement and stability. A coconut palm survives a cyclone because of flexibility; the Sydney Harbor Bridge pylons support enormous weight due to the stability of their granite blocks.

The steel columns and beams used in modern buildings are seldom solid. If we can work out which part of the beam is going to have to bear the greatest stress, it is much more economical to use hollow beams, which have strength mainly in those places. Exactly the same rules apply to columns as cables, except cables are in tension and columns under compression. Therefore the important measurement when considering how much weight a column can carry is how think it is, i.e. the area of the column. Following are columns commonly used in construction: 

1. Area = 8 X 10 = 8 sq.cm.
2. Area = 8 X 10 = 8 sq.cm = 1 mm = 0.1cm
   9.8 cm X 7.8 cm = 76.44 sq.cm
                    80 – 76.44 = 3.56 sq.cm.
3. Area = 10 X 8 = 80 sq.cm.
   80 sq.cm. – 7.9 X 8 = 78.42 sq.cm.
   80 – 78.42 = 1.58 sq.cm
4. Area = 0.1 X 8 cm X 2
   = 1.6 + 9.8 X 0.1 cm
   = 1.6 + 0.98
                 = 2.58 sq.cm.

The ultimate stress of steel is 45kN per square centimeter; therefore if column D is made of steel, the absolute weight it can carry is 2.58 X 45 = 116.1kN.

3. DOME. This is most often a hemisphere covering a space in a single piece. We see it in the human skull, which incidentally has the most perfect dome of any vertebrate; not surprising I suppose, as this form is a mirror of the heavens. The egg is the dome exemplified in the natural world (really two domes joined at the bases).

The Pantheon in Rome is a shallow dome, with increasingly lighter materials near the top, like pumice. Islam is the true home of the dome; most of Australia’s war memorials reflect this influence from the WWI campaigns in Palestine, Gallipoli and other Muslim regions.

While Western Christianity preferred spires, their Eastern counterparts communicated with their god through the symbol of heaven itself, the dome. This was so from Russia, and its onion domes, down to the giant golden dome of Sancta Sophia in Istanbul. Somewhere in between we have Rome, where the dome of St. Peters, designed by Michelangelo, was innovatively supported from sideways thrust by being circles in a huge chain secreted in the cavity. One of the most moving impressions is of a new baby’s head viewed from the top. Here we see two interpenetrating domes – just like a plan view of Rudolf Steiner’s First Goethean. The following is a student’s drawings of how this divine form gradually, over the whole of like, is fashioned by the emerging Ego into a cube – of a sort. After about age 63 (9 X 7), the human head slowly reverts to a dome form again in order to reflect its approaching supersensible destination.

4. ARCH. This word means ‘first’, and is a multitude of hemispherical 2- or 3-demensional tubular supports. The floating ribs in the human thorax is a supporting arch structure, as ins a limestone cave in nature.

The arch can be a catenary curve; the catenary equation is y = a cosh (x/a) if a is a constant distance between the lowest point of the curve and a reference axis. If a = 1, the equation is that of the hyperbolic cosine (cosh). The catenary can be generated as the locus of the focus of a parabola rolling along a straight line. A Catenoidis the surface generated when a catenary rotates about its axis, like a skipping rope.

5. LINTEL. Here we have a wide variety of horizontal supporting members, the human pelvis being a perfect – if complex but not so obvious – example. A log fallen across a ravine is a lintel, especially if holding up more than just is own weight, like a bear walking across.

6. FORCE. The first verity we must establish in engineering or mechanics is to distinguish between weight and mass. Mass is a measure of the quantity of matter in a body; it is not the matter itself, rather an abstraction based on a convention (eg., metric). Weight rather is produced by the force of gravity pulling on that mass.

In engineering we are mainly concerned with forces; wherever a force is applied, it causes another force to come into action, equal in value but opposite in direction. We can envisage these forces in eternal struggle throughout the physical world. Mass is the bailiwick of Lucifer, weight of Ahriman. This invisible adversarial principle can be illustrated thus:

by upward forces in the two supports, thus: w = R1 + R2. In a system of balance, upward forces are equal to downward forces. Forces pulling to the right are equal to forces pulling to the left. It follows therefore that the total clockwise turning effect = the total anticlockwise turning effect. A turning effect is called a Moment, and is the force X the distance from the fulcrum or support.

Whenever a body has a uniform mass (or weight), we can consider tis weight as coming entirely from its center (center of gravity. We take the moments about A, i.e. A becomes clockwise moment = 10 X 2 + 40 X 4. Anticlockwise moment = R2 X 6. Since there is equilibrium, 10 X 2 = 40 X 4 = R2 X 6. ^R2 = 20 + 160 = 180; therefore R2 = 180/6 = 30.

Taking moments about B (i.e. with B as the fulcrum), clockwise moment; R1 X 6. Anticlockwise moment; 10 X 4 ++ 40 X 2, therefore 6R1 = 10 X 4 + 40 X 2 = 40 + 80, therefore 6R1 = 120/6; R1 = 20

We can see that this answer is most likely correct because we know that upward forces must equal downward forces, i.e.

R1+R2 = 10 + 40
20 + 30 = 10 + 40

Find the value of R2: R2 X 6 = 10 X 2 + 40 X 4 = 20 + 160 = 180
                                      R2 X 6 = 180
                                      R2 = 180/6
                                      R2 = 30

With a 2^nd Class lever, we have a beam length 2x meters and a uniform weight (w) rests on a support at one end and is held in balance by an upward force (F) at the other end:

Formula: Clockwise moment: F X 2x. Anticlockwise moment: w X x,
therefore 2xF = wx.
Example: 2xF = wx, 20 X F = 100, therefore F = 5 (The convention is for weights to be shown in capital letters, distances in lower case.)

7. BRACE. This is a diagonal member supporting and joining a horizontal to a vertical member. The human clavicle (key) joins the upright sternum to the horizontal shoulder. A buttress root on a rainforest tree is a very fine brace indeed.

TENSION

1. CABLE. There are many kinds of these flexible connecting members in architecture. A clear cable in human anatomy, the first of our muscle (as compared with the previous compression-skeletal_ examples, is the biceps. These again embrace both tension and compression in their two extreme positions of arm extended (tension) and bent (compression). A hanging or taught vine is a good example in nature.

2. HANGER. When a weight is perpendicularly supported by one of more elements, such as ropes or rods (flexible or rigid), the force of gravity is compensated by an equal and opposite force. The scrotum is a hanger suspending the testes – a bunch of bananas by its stalk.

3. CANTILEVER. We see this unsupported (directly from underneath at least) member capable of holding up great weights. Our deltoid muscle works as a cantilever when our arm is extended, a simple tree branch extending out from its truck is also a cantilever

The principle of the cantilever depends on all support being at one end; it is a 3^rd Class Lever – fulcrum-effort-resistance:

It is used widely in engineering where it is difficult to support from beneath, or where a clean design line dictates no messy structure. The system can be very strong, with common materials being timber and steel (the latter often enclosed by concrete, where it is usually pre-stressed).

Now a cantilever calculation: A beam of length 3.5 meters, and of uniform weight of 0.26 kilonewtons, is hung as a cantilever. What is the bending moment at the fulcrum end )in Newton meters – Nm)?

½ X 3.5 = 1.75 meters. 0.26 kilonewtons = 260 Newtons.
1.75 X 260 = 455 Nm.

4 SHEET. This is a 2-dimensional flexible or rigid cladding or skin joining many structural members into one integral element. A sheet can be also merely non-supporting fill, as in window glass. The human trapezius is a sheet, as is a bat’s wing.

 

5. TUBE. This is a hollow, self-supporting fully enclosed wall, more often than not circular in section. Our alimentary canal, as well as every blood vessel in our body, are tubes. Bamboo is a structurally impressive example from the natural world.

6. WEB. Finally we have one of the most common structural principles in building, a simple 2-dimensional flexible (mat) or rigid (lattice) framework. Human heart muscle is a web, as is the woven bark of a palm tree. The web is often the scaffold over which the sheet is stretched.

All drawings of the 12 Principles of Structure are from students’ Architecture-Engineering lesson books. There are Five stress problems of bridge building. For instance concrete can have crushing strength of 30N per sq. meter (4500 pounds per sq. inch!)

There is a temptation to spend lots of time in this unit on the history of architecture, fascinating as it is. This must be resisted, as this area is well covered in the five years of Art History middle lessons (see my book Birth of Venus); and the many Visual Arts lessons generally.

There are also many aspects of architecture as an art and cultural expression introduced in both the Australian and World History units throughout the five years of high school. Then there is a unit dealing especially with the evolution of the built environment, the Space and Shape maths lesson in Class 8 (see What is X? Why is Y?). This unit is more concerned with the mathematics of Architecture, as they express through engineering principles; we begin here with a simple balance equation, the First-Class Lever – like a see-saw – as would have been taught in primary school (though not as early as First Class!). Quick revision exercises should be done. If three values are known, what is the 4^th? And so on.

In fact this sublime building is a perfect case study for Class 12 students to describe the whole process of designing and constructing a building; from the very first site decision to the official opening.

And while on site decisions, as a conclusion to this last maths lesson in 12 years of spiritual enlightenment – er, I mean Steiner Education -, there is evidence that the Goethean is special not only structurally, but in location as well. Apocryphal stories abound that this particular site, in the foothills of the stately Jura Mountains, was the original home of the Grail Castle in that great spiritual epic, Parsifal.

 This is a special region in all senses, geological, political and spiritual, being a region where three countries, France, Switzerland and Germany, meet, as do three mountain ranges, the Black Forest, Vosges, and the Jura.

The Grail story takes place in the 9^th Century, as a culmination of the New Age of Solomon, a time of Temple Mysteries, or Gnosticism. The Goethean is the New Temple or ‘castle’, a new-age Guardian of the Grail. The building sits up on the Dornach Hill kike a third – Ego – eye, between the busy ‘face’ of Basle below, and the Jura ‘skull’ above.

Actually there are two centers, the Goethean and nearby Arlesheim, established as an Anthroposophical center of healing. In the Grail story too, there is this dichotomy, with the Grail Castle adjacent to the Hermitage of Trevrient, a place of sustenance and teaching. Parsifal received this instruction here, and mourning Sigune was sustained by visitations from the Castle nearby.

In relation to the site, Rudolf Steiner said the one day prior to the laying of the Foundation Stone in 1913: “Karma has indicated this place so clearly, that I would not dare any longer to offer any other advice than to build here. I must say that every day fresh spiritual reasons become evident to me, which show the place that has been pressed upon us as the right one.” And next day: “We stand at this moment, led by Karma, at the place through which have passed important spiritual streams. Let us feel this evening the earnestness of the situation.”

Steiner has been heard in conversation to assert that the scene in Parsifal of Sigune with her dead bridegroom – Pieta-like across her knees – happened historically in the neighborhood of the Goethean; near Arlesheim actually. As in all myths and legends, this story describes a complex combination of both temporal and metaphysical events.

Steiner himself certainly represents the very modest model of a modern Grail Knight; a Champion of the Spirit indeed! Perhaps the conflagration which razed that most wonderful of all 20^th Century buildings, not long before its creator’s own departure, was due to the fact that the Gods of Karma decreed that on-one else was worthy to preside over it?

Who knows, with an Education of the Spirit such as described in this book, and its 23 companions in the Spiritual Syllabus Series – a 12-year Quest indeed! – one of your own students may rise – in the Parsifal sense – to ask the right quest-ion. They may even get the right answer.