Find the maximum cardinality of a set of disjoint figure-eights in the plane, and prove that it is the maximum.
Let a = 1/n be the reciprocal of a positive integer n. Let A and B be two points of the plane such that the segment AB has length 1. Prove that every continuous curve joining A to B has a chord parallel to AB and of length a. Show that if a is not the reciprocal of an integer, then there is a continuous curve joining A to B which has no such chord of length a. [Source: Challenging Mathematical Problems with Elementary Solutions vol.2]
You can get an injection from any collection of disjoint figure-eights on the plane to Q4 by picking any rational point in the interior of each of the two circles making up the figure, so there can only be countably many such disjoint figures. This is an injection because if another figure-eight had the same rational points as its representative, then it’s easy to show that the two figures must intersect somewhere.
By a figure-eight I mean a curve consisting of two topological circles that meet at a single point, e.g. Then you want to find the maximal cardinality of a set of these, such that no two intersect. Obviously there can be at least countably many; the question is whether there can be more.
Find the maximum cardinality of a set of disjoint figure-eights in the plane, and prove that it is the maximum.
Let a = 1/n be the reciprocal of a positive integer n. Let A and B be two points of the plane such that the segment AB has length 1. Prove that every continuous curve joining A to B has a chord parallel to AB and of length a. Show that if a is not the reciprocal of an integer, then there is a continuous curve joining A to B which has no such chord of length a. [Source: Challenging Mathematical Problems with Elementary Solutions vol.2]
I forgot about this one. For the first problem:
You can get an injection from any collection of disjoint figure-eights on the plane to Q4 by picking any rational point in the interior of each of the two circles making up the figure, so there can only be countably many such disjoint figures. This is an injection because if another figure-eight had the same rational points as its representative, then it’s easy to show that the two figures must intersect somewhere.
I don’t understand your first question. Can you clarify?
By a figure-eight I mean a curve consisting of two topological circles that meet at a single point, e.g. Then you want to find the maximal cardinality of a set of these, such that no two intersect. Obviously there can be at least countably many; the question is whether there can be more.
Ah, I understand. Thanks.