2. Families of Four-Point Touching Squares

This Demonstration shows four-point touching families of squares, where . The parameter determines the point by , where is the radius vector of a point . This generalizes [1], where . In this Demonstration you can vary within an appropriate range to maintain four-point touching in a family with .


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A family is said to be -touching provided each element of has a nonempty intersection with precisely other elements of . A -touching family is said to be point--touching if any two touching sets have only a single common point [2].
Grünbaum has shown that example (with dihedral symmetry) in [3] is impossible. He has also said that it is not clear whether there are any point-4-touching families with fewer than 48 squares.
[1] I. Hafner. "Families of Four-Point Touching Squares" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/FamiliesOfFourPointTouchingSquares.
[2] B. Grünbaum, "Families of Point-Touching Squares," Geombinatorics, 12(4), 2003 pp. 167–174. (Mar 27, 2019) sites.math.washington.edu/~grunbaum/Familiesofpointtouchin.pdf.
[3] E. Friedman, "Squares Touching a Constant Number of Other Squares," Geombinatorics, 12(2), 2002 pp. 55–60.
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