Cayley’s Sextic Essay, Research Paper
The curve, Cayley?s Sextic can be described by the Cartesian equation: 4(x^2 + y^2 ? ax)^3 = 27a^2(x^2 + y^2)^2. It is the involute of a nephroiod curve because of its slight kidney shape and because they are parallel curves. This curve was first discovered by a mathematician by the name of Colin Maclaurin. Maclaurin who was born in February of 1698, became a student at Glasgow University in Scotland during his early teen years. It was here that he discovered his abilities in mathematics and began working towards a future in geometry and mathematics. In 1717 Maclaurin was given the job as the professor of mathematics at Marischal College in the University of Aberdeen. Later during his mathematical career, Maclaurin wrote Geometrica Organica, a book which displayed early ideas of what later becomes known as the curve, Cayley?s Sextic.
The actual man credited with the distinct discovery of Cayley?s Sextic is the man it is named after, Arthur Cayley. Cayley, who had a family of English ancestry, lived in St. Petersburg, Russia during his childhood where he attended his first years of schooling. In 1835 he began attending King?s College School in England because of his promise as a mathematician. After Cayley became a lawyer and studied math during his spare time, publishing papers in various mathematical journals. These journals were later looked at by Archibald and in a paper published in 1900 in Strasbourg he gave Cayley the honor of having the curve named after him.
Cayley?s Sextic
The polar form of the equation for the curve, Cayley?s Sextic, is shown as:
r = 4a cos^3 (q/3). For the specific equation for the graph, the polar form is the equation of greatest ease of use. Use 1 in place of ?a? and switch the calculator to polar form. The best viewing window for this graph is q min= -360; q max= 360; q step= 10; x-min= -5; x-max= 5; x scale= 1; y-min= -5; y-max= 5; y scale= 1. This window and equation will give an excellent picture of the curve, Cayley?s Sextic.
When ?a? is increased in the equation for the curve, the entire curve increases in size, giving it a larger area. The value for ?x? is greatly increased on the right side positive y-axis, while the value for ?x? on the left side negative y-axis becomes gradually more negative at a much lower rate then that of the right side positive y-axis. The ?y? values for the curve increase and decrease at the same rate on both sides of the x axis when the value of ?a? changes.
When the value of ?a? becomes negative, the curve is flipped over the y-axis. When the value of ?a? decreases to a lower negative number the area of the curve increases giving it a larger area. The value for ?x? in greatly increased on the left side, negative y-axis, while the ?x? on the right side positive y-axis becomes gradually more positive at a much lower rate then that of the left side negative y-axis. The ?y? values once again increase and decrease at the same rate on both sides of the x-axis when the value of ?a? changes.