I have solved the Erdos-Ulam problem (see link) and can construct a set that satisfies the conditions (dense in R2 with all interpoint distances rational). I have expanded the solution from two dimensions to n dimensions for any integer n. I would welcome any comments.
1 Answer
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The error is in this sentence:
As a circle with its centre on the Base Plane, $T_{n-1}$ will have at least two points on the Base Plane.
Circles in an ambient space of more than three dimensions don't intersect every plane passing through their center. For instance, consider in $4$-space the circle given by intersecting the $zw$ plane with the sphere of radius $1$; it does not pass through any points in the $xy$ plane.
