Sliceform Craters: An Exploration in Equations

Sliceform Craters: An Exploration in EquationsAuthor(s): John SharpSource: Mathematics in School, Vol. 33, No. 2 (Mar., 2004), pp. 12-21Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215677 .

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Sliceform Craters

an exploration in equations by John Sharp

Although you can teach many aspects of mathematics, it is up to the student to learn. Small children do not need teachers so much as the ability to learn for themselves by play and exploration. Such ways of learning should not stop later in life and playing with equations and being creative with them is a good way to learn. This exploration shows some techniques for creating new surfaces as Sliceforms by 'playing' with equations and getting a feel for how they behave.

The 'Witch' of Agnesi Maria Gaetana Agnesi (1718-1799) was an extraordinary woman, who was a linguist, philosopher and took over from her father as professor of mathematics at the University of Bologna. She published her Institutuzioni Analitiche in 1748 which was translated into French and English. It contains the description of a curve which is still associated with her name as the Witch of Agnesi. The name Witch arose because of a strange set of misprints and mistranslations. It was originally called versoria which means. free to move in any direction. Agnesi's book printed it as versiera which means hobgoblin or ghost or 'spirit of the night'. When the book was translated into English by John Colson he called it 'Witch', a name that has stuck. Perhaps he saw it as the path of a witch's broomstick as she flew through the night.

To draw the witch (Fig. 1), take a circle and its diameter with a fixed line perpendicular to the diameter. Rotate a line through A to cut the circle at Q and the line at B. Draw perpendiculars to the line at B and to the diameter through Q to intersect at P The locus of P is the Witch.

Q P

A

Fig. 1

12

With A as the origin, the equation of the witch is a3

(X2 + a2) (equation A)

The plotted curve is shown in Figure 2 and you can see that it is symmetrical about they-axis and asymptotic to the x-axis.

Fig. 2

The following exploration takes the equation of the curve and modifies it to show how to create a number of Sliceform models as surfaces of revolution with the slices plotted as equations.

Surfaces of Revolution

To convert the curve to a surface of revolution, it needs to be considered as a curve in a plane rotating about the z-axis. Figure 3 shows the curve rotated about its central vertical axis.

z

P y

A x

B O

Fig. 3

Consider the point P of the curve and its projection onto the xy-plane at point A and consider the plane in which the curve lies. Compare distances/coordinates with the ones used to plot the curve in the usual cartesian xy-plane in

Mathematics in School, March 2004 The MA web site www.m-a.org.uk



Fig. 4

(Note: Pairs of images like this are stereoscopic pairs which enable you to see the surface in three dimensions. Place the page 8-9 inches from your eyes; then cross your eyes slightly so that you see three images. Focus your attention on the centre one which should appear as a three

dimensional image. This may take a few seconds, and once you can do it with one image, you should find it easier with others.)

equation A. The distance OA corresponds to the original x- coordinate in equation A and the distance AP to the original y-coordinate. The height AP is the z-coordinate of the curve in space, and OA can be obtained from Pythagoras's theorem:

OA2 = x2 + y2 (equation B)

where x and y are the coordinates of the equation of the surface. The surface equation defining the point P is then obtained by substitution using equation A, remembering that the x andy in the two equations do not match. This is a good exercise in algebra and matching different coordinate systems. For the surface, z is equal to AP which is equivalent to the y-value in equation A, with the x-value of equation A being equivalent to OA in equation B. So the surface equation becomes:

a3 S= (X2 a y2 + a2)

(equation C)

and the surface appears as in Figure 4.

Creating the Sliceform Model

To plot the slices for the surface, an imaginary grid is placed on the xy-plane and the slices defined as planes perpendicular to this grid. The slices in each direction are then plots of z against x (with y kept constant) and z against y (with x kept constant). The constant values in each case are defined by a line on the grid.

However, because the surface has rotational symmetry and hence some mirror symmetry, only a limited number of slices need to be plotted. The 9 by 9 grid shown in Figure 5 has some equivalent slices marked with arrows. The only slices that need to be plotted are the central one and the four to one side of it.

Fig. 5

In order to give some support to the surface, a base has to be added and the slots to fit each set of slices together also need to be added. Although the slices in each direction are plotted with the same curve, the slots have to be in different directions. Figure 6 shows a set of unique shapes of slices from each direction. In addition to these, you will need


another set of each of the slices which are not the central ones (the left-hand ones).

Fig. 6

Assembly instructions

Having designed the slices, copied them onto card and cut them out, it is important to cut slots by cutting twice (either side of the slot lines) and then removing the hair-like piece, so that the slice in the other direction fits the thickness of the card. If you do not do this, the model will buckle as the forces try to even themselves out. When assembling the model, start from the centre ones and add pairs either side of each centre.

Modifying the Equation to make 'Craters'

Such equations make interesting Sliceforms, but they can be enhanced to yield even better Sliceforms. The following method creates surfaces which look like craters.

Figure 7 shows a modification of the curve of equation A which shifts the peak in the x-direction by plotting the value (x-d) instead ofx where d is a constant (and having the value of 0.8 in Figure 7, with a being 1), so that the equation becomes:

a3

((X- d)2+ a2) (equation D)

Fig. 7

Now if we modify equation D by using the absolute value of x (that is ignore any sign of x when calculating y), then we can get a symmetric curve with two peaks. The equation becomes:

a3

(( x - d)2 + a2) (equation E)

13



Fig. 8

As Figure 8 shows, increasing the value of d produces a curve with two separate humps and a lower dip in the middle. The value of a is 1 in all cases and d has values of 1 at the top, then 2 and 3.

In order to convert to a surface, since OA =~ (x2 + y2) the equivalent of (I x - d)2 has to be found by squaring the bracket contents before converting, that is x2 -2 Ix |d + d2 so the surface equation becomes:

a3 Z =((X2 +y2- 2d I '(x2 +y2)l +d2)+ a)

(equation F)

and the surface appears as in Figure 9 if a has the value 1 and d has the value 2:

This gives craters which are rounded. However, if the absolute value is made to be the difference of x and d before squaring, that is I x2 -d I instead of ( Ix I-d)2 to give equation E

a3 Y(IX2 - dl +a2) (equation G)

Fig. 9

Fig. 11

14

Fig. 10

then the edges of the crater are much sharper as shown in Figure 10.

The equation of the surface of revolution then becomes:

a3 (Z ((x2 +y2-_d) I +a2) (equation H)

and the surface when plotted with a equal to 0.85 and d equal to 3 is shown in Figure 11.

Because of the way the surface plotting program creates the surface as a set of lines, the appearance is similar to the way the Sliceform model will appear, albeit with more slices than it would be practical to make.

The Sliceform Models

Making the model uses the same technique as previously, just changing the equation of the surface. The model of the crater with the rounded edges is shown in Figure 12, the one with sharper edges in Figure 13 and that of the original surface of revolution in Figure 14 for comparison.




Witch surface of

revolution page 1

C John Sharp 2003

IMPORTANT Please read the assembly instructions on page 13.

Mathematics in School, March 2004 The MA web site www.m-a.org.uk 15



Witch surface of

revolution page 2

@ John Sharp 2003

16 Mathematics in School, March 2004 The MA web site www.m-a.org.uk



Smooth Crater

page 1

C John Sharp 2003





Smooth Crater

page 2

@ John Sharp 2003




Jagged Crater

page 1

C John Sharp 2003





Jagged Crater

page 2

@ John Sharp 2003




I

I

I

Fig. 12

4 p

Fig. 13

~ ~i~-;;--;~-;-- ;-;-;A

x;

Fig. 14

References As part of designing models, it is a good idea to experiment on the computer. The ideas above were plotted with a free set of programs for plotting surfaces called Plot3D by Clark Dailey, with data entered through a dialog box and allowing very fine control over the plotting. Available at www.simtel.net and also on CD from Simtel. Download the files from the Win98 section / Graphics / Misc. Graphic Programs & Utilities There are three programs in the suite. Plot3DF3.ZIP - Plot F(x,y,z) = 0 as 3D surface. Plot3DP1.ZIP - Plots a 3D surface parametrically. Plot3DZ3.ZIP - Plot z = f(x,y) as 3D surface

There are two books on Sliceforms:

John Sharp Sliceforms. Mathematical models from paper sections, including eight models of surfaces to cut out and construct. Tarquin Publications.

John Sharp Surfaces: Explorations with Sliceforms. This is a comprehensive book on the geometry of surfaces with many examples and ideas for creating new surfaces at all levels. QED Books.

Keywords: Sliceforms; Equations.

Author John Sharp, 20 The Glebe, Watford WD25 OLR.

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Sliceform Craters: An Exploration in Equations

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