Mathematician Helaman Ferguson combines science

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1 L A B 19 MATHEMATICAL SCULPTURES Parametric Srfaces Mathematician Helaman Fergson combines science and art with his niqe mathematical sclptres. Fergson s sclptres are the concrete embodiments of mathematical concepts. His sclptres incorporate ideas sch as series expansions and vector fields into their creation. Some of the basic images in his work are tori and doble tori, Möbis strips, and trefoil knots. Observations One example of Helaman Fergson s work is Umbilic Tors NC. The work, which stands 27 inches high, was created by the maniplation of matrices associated with homogeneos cbic polynomials in two variables. With only one edge, yo can trace the edge of Umbilic Tors NC three times arond before retrning to the starting point. Prpose In this lab, yo will analyze Helaman Fergson s work Umbilic Tors NC. Yo will se Mathcad to aid yo in yor analysis. References For more information abot mathematical sclptres see Helaman Fergson, Mathematics in Stone and Bronze from Meridian Creative Grop. Hoghton Mifflin Company. All rights reserved. LAB 19 MATHEMATICAL SCULPTURES 1

2 Data Helaman Fergson s work Umbilic Tors NC is shown below. This form can be written as a parametric srface sing the following set of parametric eqations. x sin 7 cos 3 2v 2 cos 3 v y cos 7 cos 3 2v 2 cos 3 v z sin 3 2v 2 sin 3 v, v A graph of the parametric eqations is stored in the Mathcad file called LAB19.MCD. Hoghton Mifflin Company. All rights reserved. 2 Mathcad Lab Manal for Calcls

3 Exercises Name Date Class Instrctor 1. Interchanging Eqations. The view of the tors in this lab s Data is along the z-axis. Relative to the graph of the tors given in this lab s Data, describe the graph of the tors given by the parametric eqations x cos 7 cos 3 2v 2 cos 3 v y sin 7 cos 3 2v 2 cos 3 v z sin 3 2v 2 sin 3 v v. with and Use Mathcad to confirm yor description. Hoghton Mifflin Company. All rights reserved. 2. Interchanging Eqations, Part II. Repeat Exercise 1 for the following set of eqations with and x sin 3 2v 2 sin 3 v v. y cos 7 cos 3 2v 2 cos 3 v z sin 7 cos 3 2v 2 cos 3 v LAB 19 MATHEMATICAL SCULPTURES 3

4 3. Interchanging Eqations, Part III. Repeat Exercise 1 for the following set of eqations with and y sin 3 2v 2 sin 3 v v. x sin 7 cos 3 2v 2 cos 3 v z cos 7 cos 3 2v 2 cos 3 v 4. An Elliptical Tors. When viewed along the z-axis, the tors appears nearly circlar. Is it possible to alter this appearance so that the shape of the tors is more like that of an elongated ellipse? If so, explain how this cold be done and se Mathcad to graph an elliptical tors. 5. Comparing Graphs. Explain how the mbilic tors shown in this lab s Data is similar to the Möbis strip shown below. Hoghton Mifflin Company. All rights reserved. 4 Mathcad Lab Manal for Calcls

5 6. Graphing Portions of the Tors. Use Mathcad to recreate each of the following graphs. Explain yor strategy to recreate each graph. Also, write the parametric eqations and - and v-ranges yo sed to recreate each graph. Hoghton Mifflin Company. All rights reserved. LAB 19 MATHEMATICAL SCULPTURES 5

6 7. Cross Section. An example of a cross-section of Umbilic Tors NC is shown below. The cross-section is a hypocycloid with three arches. A hypocycloid is a crve generated by a point on the circmference of a circle rolling within the circmference of a larger circle. The parametric eqations of a hypocycloid in the xy-plane are x a b cos b cos a b b y a b sin b sin a b b where 0 2. One possible set of parametric eqations of the hypocycloid shown above is as follows. x 2 cos cos 2 y 2 cos cos 2 Use Mathcad to graph a hypocycloid with for arches and a hypocycloid with five arches. What vales of a and b did yo se to create yor graphs? What is the relationship between a and b for a hypocycloid with three arches? What is the relationship between a and b for a hypocycloid with for arches? What is the relationship between a and b for a hypocycloid with n arches? Hoghton Mifflin Company. All rights reserved. 6 Mathcad Lab Manal for Calcls