© ATM 2010 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions. HOW TO ANAMORPH John Sharp shows how you can learn in many different ways with anamorphosis. The centre spread on Anamorphic Art in MT199 ‘see’ in the sense of understanding, but in the sense contains some interesting comments by Julian of looking. The practical aspect of having to solve Beever: “I hate maths. I don’t use maths, but I do your own problems is a very important part of use perspective.” These curious statements reflect learning. Not enough is done to teach how to learn. comments students used to make to me when I was There is one other aspect of art which can be very teaching art and geometry in Adult education. They powerful in teaching mathematics: grasping the would say that they enjoyed geometry in school and 3-D world. This is often neglected since most art were good at it, but disliked maths and were no taught with mathematics only deals with the plane, good at it. Both ideas highlight the image mathe- usually through symmetry and pattern. matics has for the average person. Art uses mathe- My main reason for writing this article, however, matics in many ways, but in none of them is is to try to unravel figures 1 and 2 of Colin Foster’s perspective more part of it or common to our Anamorphic Art article in MT199. I looked at these culture. My teaching and my involvement with figures and thought there is something not quite mathematics and art in the Bridges Conference right. They reminded me of one of my favourite connecting the two (see MT193 p.23) have quotes “I have yet to see any problem, however convinced me more and more that concepts for complicated, which when looked at in the right way understanding mathematics can be achieved by the did not become still more complicated”. This is use of art. When teaching, I found that I could allied to the only razor I use, Occam’s razor, which cover quite deep mathematics with a wide range of says that the simplest solution to a problem is the students without telling them that is what I was most likely. In order to explain why Colin may doing and they had no problems in understanding it. appear to be simplifying the problem but has One reason I believe in teaching mathematics complicated it no end, I need to show how through art is that it gives ownership to learners. By anamorphic images are constructed and show how this I mean that learners are creating their own they differ from perspective. His method is mathe- work and they have to be able to express themselves matically valid, but not the easiest to work with and, and produce their own result, not the answer in the to the best of my knowledge, it does not appear in back of the book. In using mathematics as a tool it any published work on anamorphosis because it becomes less intimidating. Learners can not only confuses perspective with anamorphosis. Figures 1 and 2: Colin Foster’s creation of an anamorphosis. MATHEMATICS TEACHING 217 / MARCH 2010 7 Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY © ATM 2010 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions. Figure 3: Creating a perspective picture Figure 3 shows what happens when you create a perspective picture. At the left is the three dimen- sional view, in fact a perspective picture of a making a perspective picture, and at the right an elevation or cross section from above. The picture plane is a window and you draw lines, or follow rays of light, from the eye point E to the objects you want to represent. You then mark the intersection of these lines with the picture plane. Note also that the picture plane is perpendicular to the line of sight. Now let’s look at a tiled floor in perspective as in figure 4. As with figure 3, the left part shows a perspective of a grid on the floor viewed in Figure 4: A grid in perspective perspective. At the right is the picture you would see looking through the window/picture plane. Point E is the eyepoint with EQ the line perpendi- cular to the picture plane; C is the central vanishing point and FG the image of the horizon line. In the image on the picture plane on the right, the set of lines of the grid that are parallel to the base of the picture, and the horizon line, get closer together as they correspond to the farthest ones. Everyone is familiar with how perspective makes distant objects smaller and that if you look along railway lines they converge to a point. The sleepers which physically are equal distances apart become closer together as they go into the distance in the perspective picture. Now if we look at Colin’s and my figures 1 and 2 and ignore the cube for the moment we see that he has created a perspective picture with his grids. Colin’s figure 2 is a tiled floor, where the anamor- phic picture is drawn, and figure 1 is a perspective view of it, so that it appears normal. I am puzzled by how he obtained the anamorphosis since that is a difficult geometrical procedure working in this Figure 5: Creation of an anamorphic image by projection way. 8 MATHEMATICS TEACHING 217 / MARCH 2010 Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY © ATM 2010 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions. Figure 6: Projecting a circle to an ellipse Figure 5 shows the fundamental difference in I have taught perspective, I have found that some creating an anamorphosis. You start with a people find this difficult in the way that others are predrawn image and you project it onto a plane. tone deaf. So you might find it useful to make a Because you are projecting, it is the reverse of a simple cardboard model with some string to show perspective. You start from the eyepoint E and you the lines from the eye as in figure 5, or better still join each point of the predrawn image and extend include it as part of a lesson. the lines until you reach the picture plane – the ground in Julian Beever’s pictures. Suppose, as in figure 5, we start with a square as our predrawn image. We want to look at it as if it were a normal perspective, so we look with the line of sight to the centre of the square perpendicular. Thus in the side elevation, line EC is normal to the image plane. Note how the top of the anamorphic distorted picture, the grey shape, tapers towards point A. This is the opposite of the tapering to the central vanishing point of a perspective picture, tapering to the central vanishing point as in figure 4. Note also how the eyepoint E does not sit above the point A. This projection of a image is not always easy to draw, so you often see books of perspective from a couple of centuries ago including a method which uses a candle to project an image by means of pricked holes in a piece of paper. A simpler version of creating an anamorphic circle is shown in figure 6. This is from a book of recreations for boys dated 1852. A candle is placed next to the hole at H and you then draw round it’s projected image. When you remove the candle and look through the hole you see a circle. Note how you are looking obliquely at the picture on the base plane. The same book has a complete anamorphic picture (figure 7) with a device for making the viewing hole. If you look at the image obliquely from a suitable distance, then everything stands up Creating an anamorphic grid Drawing an anamorphic grid is not too difficult if you follow figure 5. It is a good exercise in being Figure 7: A complete anamorphic picture able to think from three to two dimensions. When MATHEMATICS TEACHING 217 / MARCH 2010 9 Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY © ATM 2010 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions. It is easier to see how to create the grid if you can see the final result. So figure 8 shows the square grid and its anamorphic equivalent. Figure 9 shows the steps in creating the grid which begins with drawing the diagram that is the lower diagram of figure 5. So figure 9a is drawn by first drawing the horizontal line which is the cross section of the plane of the anamorphic grid. Then line DG is drawn to match the edge length of the square grid. Line CE is drawn perpendicular to GD through its centre and point E chosen as the eyepoint. Next EA is drawn perpendicular to EC at Figure 8: The square grid and its anamorphic equivalent E. Finally E is joined to G to give point F. This allows measurements for starting figure 9b. AF is drawn and also a point D which is where D1-D2 cuts it. A perpendicular is drawn through D and its length D1-D2 is equal to CG is figure 9a since we a) are working with a square grid.