Ten points on a cubic. (English) Zbl 1532.14063
This well-written expository paper discusses geometric and combinatorial properties of cubic curves, especially configurations of points on such curves. The authors combine this with fascinating historical asides on Blaise Pascal and other mathematical researchers in this topic. The main result they establish is a straightedge construction that checks whether 10 general points lie on a cubic.
Theorem: Given ten points in the plane, there exists a straightedge construction that produces three points such that the original ten points lie on a cubic curve if and only if the three points are collinear.
Closely related is the following result due to Pascal:
Theorem (Pascal). If six distinct points A, B, C, a, b, and c lie on a conic, then the lines Ab, Bc, and Ca meet the lines aB, bC, and cA in three new points and these new points are collinear.
The line through the three new points is called the Pascal line. The authors point out the following amazing but true
Fact: There are \(6! = 720\) ways to reorder the points, but these give rise to only \(60\) different Pascal lines. The authors delve into this arrangement of \(60\) lines, known as Pascal’s Hexagrammum Mysticum, and discuss some of its amazing combinatorial properties.
The paper closes with a series of eight problems and four research problems. For example, one such problem is:
Problem: Given five points on a conic, use Pascal’s theorem to show that you can construct the tangent line to the conic at each of the five points.
One such research problem is:
Research Problem: Given 9 points sitting on a unique cubic curve and a subset consisting of five of these points sitting on a unique conic, construct the sixth point of intersection of the cubic with the conic using a straightedge.
For further details, the reader is referred to the paper itself.
Theorem: Given ten points in the plane, there exists a straightedge construction that produces three points such that the original ten points lie on a cubic curve if and only if the three points are collinear.
Closely related is the following result due to Pascal:
Theorem (Pascal). If six distinct points A, B, C, a, b, and c lie on a conic, then the lines Ab, Bc, and Ca meet the lines aB, bC, and cA in three new points and these new points are collinear.
The line through the three new points is called the Pascal line. The authors point out the following amazing but true
Fact: There are \(6! = 720\) ways to reorder the points, but these give rise to only \(60\) different Pascal lines. The authors delve into this arrangement of \(60\) lines, known as Pascal’s Hexagrammum Mysticum, and discuss some of its amazing combinatorial properties.
The paper closes with a series of eight problems and four research problems. For example, one such problem is:
Problem: Given five points on a conic, use Pascal’s theorem to show that you can construct the tangent line to the conic at each of the five points.
One such research problem is:
Research Problem: Given 9 points sitting on a unique cubic curve and a subset consisting of five of these points sitting on a unique conic, construct the sixth point of intersection of the cubic with the conic using a straightedge.
For further details, the reader is referred to the paper itself.
Reviewer: David Joyner (Annapolis)
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