Rebuilding a Sliced Cube

The projections shown in Figure 4.2j are of a cube that is cut by a single plane face that cuts two edges of the cube in their mid- points, and one edge of the cube at a one-third point. Reconstruct

One method of constructing an accurate view of the cut triangular face in the previ- ous task is known as an auxiliary projection, which is a method of drawing a projection from a useful direction. Step-by-step instructions are given in Figure 4.2k. Notice what you have to do to make sense of the diagram and the words, by moving your attention back and forth.

(i) Produce construction lines orthogonally (at right angles) to the cut face. These ensure that the triangle you construct will have the correct height.

(ii) Drawn an axis perpendicular to the pro- jection lines.

(iii) The length of the base is taken from the appropriate view – in this case the plan. (iv) The resulting triangle will have the

dimensions of the cut face.

Historically, auxiliary projections were used to study the two-dimensional shapes that arise from sections of a cone. The curves are often referred to as conics. It is thought they were first studied systematically by Menaechmus who was a mathematics tutor to Alexander the Great (approx 350 BCE). Imagine slicing through a cone with a

single plane.The aim is to find out what two-dimensional shapes can arise. Note that the cone has to be complete: as well as the familiar conical shape there is another conical part upside down above the vertex.

The easiest case is a slice perpendicular to the axis of the cone, parallel to the base. The section will be a circle, and this is shown in two projections in Figure 4.2l. The line AB in the front view shows where the edges of the cutting plane intersect with the sides of the cone; the large circle is the ‘base’ of the cone, and the small circle is 64 BLOCK 1

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It is not as easy as it might seem at first. Where did you start when trying to sketch the cube: with a cube that you then ‘sliced’, or with the slice itself, or with the part of the solid not touched by the slice? Being aware of choices is very useful because if one approach runs into difficulty, you know you can try another approach.

Did you use dotty paper or plasticine? What properties does the triangular slice have (what kind of triangle is it, what are the ratios of its side lengths)?

Figure 4.2k M L A B A B P M L A B P A B P M P O L Figure 4.2l

the actual section slice. The circular section will form the top circular face of the resulting solid (and the bottom circular face of the part cut away).

If the slicing plane is at an angle to the vertical axis of the cone, then the cone viewed from the side is as shown at the bottom of the second sketch in Figure 4.2l. The resulting section is an ellipse. However, the plan does not give the actual dimen- sions of the ellipse, because the ellipse is not parallel to the base. Although the plan shows an ellipse, it is in fact a foreshortened version of the slice. It is possible now to use an auxiliary projection to construct the actual shape of the ellipse.

Consider the point P in the third sketch in Figure 4.2l.The plan shows two points, L and M, on the edge of the ellipse that both correspond to P in the elevation. To determine the plan view of the ellipse you need to know where L and M are, for all positions of P.

For each position of P, points L and M lie on a circle with centre O (in line with the vertex of the cone).The radius of this circle OM has the same radius as the hori- zontal circle that would be formed by a

horizontal plane cutting the plane that passes through P. Although it may take a moment or two to sort this out, it is an excellent opportu- nity to see how what at first seems difficult and strange can actually be made sense of.

Dynamic geometry software is ideal for doing this sort of geometrical reasoning, ending with the trace (or locus) of the points L and M to see the ellipse: open the interactive files ‘4c Sliced

Cube’ and ‘4d Slicing Cones’.

Many artists in the sixteenth century did this sort of work in order to master the art of realistic drawing and painting. For example, the German artist and mathematician Dürer suggested the elliptical cross-section can be found in his

Treatise on Measurement of 1525 (Figure 4.2m).

You will see more of Dürer’s contribution to mathematics and art in the next section.

4.3 EXPLORING HIDDEN GEOMETRY IN WESTERN ART

The next part of the chapter focuses on how artists learned to reduce a three-dimensional world to a canvas so that it appears real.A technique of drawing in perspective was known to the artists who decorated walls in Pompeii, but was lost as a result of the eruption of Vesuvius. It was then rediscovered and reconstructed in the Renaissance. Artists studied geometry in order to understand how to achieve realism in their paintings.

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Figure 4.2m

Reflection 4.2

More valuable than trying to master the techniques of projections is to observe the various states of confidence and loss of confidence, of smooth flow and struggle that you experienced. How can learners be helped to appreciate that some struggle, some challenge is worthwhile, because of the feeling of satisfaction it brings when finally things slot into place?

In a world of digital cameras, everyone is familiar with interpreting the depth in photographs, but it is something that is learned by young children. It is not ‘obvi- ous’ from birth. The photograph (Figure 4.3a) highlights features that provide depth cues that a young child might mis- interpret. Things farther away are smaller, and lines that are parallel in three dimen- sions are not shown parallel in the picture. You can get further understanding as to how depth perception works by studying William Hogarth’s use of the rules of per-

spective to fool the viewer (see Figure 4.3b). The picture was included in a mathematical treatise of 1754 by Brook Taylor: Method of Perspective.

One of the first to write about the rules of perspective from a mathematical viewpoint was Alberti (1404–72), in two treatises, one of which, De pictura written in Latin, was addressed to scholars while the other, Della Pittura, written in Italian, was perhaps aimed at a more general audience. In the former of these Alberti writes: ‘A paint- ing is the intersection of a visual pyramid at a given distance, with a fixed centre and a defined position of light, represented by art with lines and colours of a given

surface.’ The illustrations in Figures 4.3c and 4.3d show how the German Dürer (1471–1528) illustrated Alberti’s ideas in a number of woodcuts.

In a woodcut produced in 1525 (Figure 4.3c) the lute is shown being drawn in the plane of the wooden frame from the viewpoint of the ‘eye’ screwed into the wall. Notice how the picture has been swung out of the way so as to mark a point where the string, representing a ray of light from a point on the lute to the ‘eye’ crosses the frame. This point is therefore projected onto the plane of the picture where the string intersects the plane. In this way every point in the pic- ture of the lute could be found, somewhat tediously!

In the second woodcut (Figure 4.3d), an artist is shown keeping his eye at a set view- point, which determines where rays from his subject will intersect a vertical grid. He then transfers this information to the grid on his drawing (without moving his head!).

One of the famous examples used by Alberti

to describe his geometrical understanding of perspective is that of pavimento – pic- tures that include square tiled floors. Alberti’s methods were later simplified by Piero della Francesca (1416–92) who was born near Florence. Della Francesca combined his skills in mathematics and art to write a number of treatises about how to achieve 66 BLOCK 1

Figure 4.3a

Figure 4.3b

Figure 4.3c

depth in paintings and drawings. One of these, On Perspective is written very much as a book about geometry, laid out in the style of Euclid’s Elements with theorems and proofs.

Piero articulated what Dürer later illustrated:

First is sight, that is to say the eye; second is the form of the thing seen; third is the distance from the eye to the thing seen; fourth are the lines which leave the boundaries of the object and come to the eye; fifth is the intersection, which comes between the eye and the thing seen, and on which it is intended to record the object.

Getting a tiled floor to look right is no mean feat, because the proportions need to be just right (and the same technique enables objects to be drawn at heights appropriate to their distance away in the picture). Drawing tiled floors became a mark of an accomplished and skilled artist, until techniques were developed for doing it geomet- rically. This section ends with a description of how it can be done and an interactive file ‘4e Vanishing Point’ so that you can experiment and appreciate the sophisticated reasoning involved.

The technique for drawing in perspective makes use of two observations:

Parallel lines are depicted as lines passing through a common point called the vanish- ing point for that family.

Vanishing points for different families of parallel lines all lie on a single line in the pic- ture called the horizon.

Figure 4.3e shows a chessboard-style grid.

The interactive file ‘4e Vanishing Point’ makes it possible to explore how a grid might appear if viewed from different viewpoints.

Using a vanishing point is a good start, but there remains the problem of spacing. How are the sizes of the tiles chosen so as to produce an appropriate sense of fore- shortening? Some step-by-step advice about how to construct a tiled pavement is given in Figure 4.3f overleaf. Apart from the edges of the tiles that form two families of parallel lines, there is a third family: the main diagonals (and, of course, other diago- nal families as well). Each family will have its own vanishing point, and the relative positions of these will determine the foreshortening effect.

Altering the positions of V and T in the interactive file ‘4e Vanishing Point’ affords insight into various effects that can be achieved. What is at first surprising, and must have excited artists developing the ideas, is that all the other diagonal families are forced to meet at their own vanishing points, and that these all lie on the horizon.

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In the next task, you are asked to follow instructions to construct a pavement from a given viewing position.

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(i) A base line with five points equally spaced

(ii) Line projected from each of the points on the base line passing through a single point – the vanishing point. (This is because in the grid these lines are parallel and at right angles to the plane of the page.)

(iii) Draw the diagonal from the front left corner of the grid to pass through T, a point on the horizon: the horizontal line passing through the vanishing point V.

(iv) Where this diagonal line cuts the verti- cal line through the front right corner of the grid identifies the distance of the back edge of the grid (shown by the additional horizontal line here).

(v) Where the diagonal from front right to back left of the grid cuts the grid lines that converge at the vanishing point defines the positions of the horizontal transversals.

(vi) The grid can now be constructed.

Figure 4.3f