I’ve taken an interest in steepled arrangements of quadrilaterals; i.e., an arrangement of n quadrilaterals with 2n vertices such that the intersection of any two quadrilaterals is either a vertex or the empty set, and each quadrilateral meets four others at its vertices. The implication is that n >=5, and I’m focused on the 3 dimensional case. A link from an earlier open thread shows that such an arrangement is possible.
The term steepled refers to a hand position where the corresponding fingers of each hand meet at the fingertips, forming a ‘steeple’.
Consider an ant crawling on the surface of one of these arrangements. Starting at a vertex, she makes a bee-line (ant-line?) for the diagonally opposite corner of the quadrilateral she’s on, and then enters the next quadrilateral which intersects at that point, and repeats the process again and again, always going across the diagonal. Eventually, she must arrive at her starting point, because there are a finite number of vertices. She may not have hit all the vertices, but she did not retrace her steps at any point.
Given a numerical labeling of the quadrilaterals, her path can be represented by a cycle of numerals corresponding to each quadrilateral she traversed. If there are unvisited vertices, the process can be repeated starting at an unvisited vertex, generating another cycle, until all vertices have been visited.
Different cycles or product of cycles (different even after a permutation of quadrilateral labels) represent different steepled quadrilateral arrangement types. The cycles have the following properties:
Each of the n numerals appears exactly twice (corresponding to the two diagonals of each quadrilateral).
Each numeral is neighbored in the cycle(s) by four different numerals between its two appearances.
Each cycle must have length >= 3.
Looking at the case n = 5, there are at least 7 distinct potential quadrilateral arrangements—i.e., 7 different eligible cycle products of the numerals 0 through 4. I’m looking into the question of whether each of these represents a physically possible 3-d steepled quadrilateral arrangement, and if so, can it be accomplished with all convex quadrilaterals or not. (That is, are you forced to use any non-convex quadrilaterals to construct the arrangement.)
I’m planning on posting this as a question to Math StackExchange, but I prefer to first be confident I can answer the question asked within a month—due to the logistics of that site, it’s possible for a question to disappear after a month if no-one has answered it.
I’ve taken an interest in steepled arrangements of quadrilaterals; i.e., an arrangement of n quadrilaterals with 2n vertices such that the intersection of any two quadrilaterals is either a vertex or the empty set, and each quadrilateral meets four others at its vertices. The implication is that n >=5, and I’m focused on the 3 dimensional case. A link from an earlier open thread shows that such an arrangement is possible.
The term steepled refers to a hand position where the corresponding fingers of each hand meet at the fingertips, forming a ‘steeple’.
Consider an ant crawling on the surface of one of these arrangements. Starting at a vertex, she makes a bee-line (ant-line?) for the diagonally opposite corner of the quadrilateral she’s on, and then enters the next quadrilateral which intersects at that point, and repeats the process again and again, always going across the diagonal. Eventually, she must arrive at her starting point, because there are a finite number of vertices. She may not have hit all the vertices, but she did not retrace her steps at any point.
Given a numerical labeling of the quadrilaterals, her path can be represented by a cycle of numerals corresponding to each quadrilateral she traversed. If there are unvisited vertices, the process can be repeated starting at an unvisited vertex, generating another cycle, until all vertices have been visited.
Different cycles or product of cycles (different even after a permutation of quadrilateral labels) represent different steepled quadrilateral arrangement types. The cycles have the following properties:
Each of the n numerals appears exactly twice (corresponding to the two diagonals of each quadrilateral).
Each numeral is neighbored in the cycle(s) by four different numerals between its two appearances.
Each cycle must have length >= 3.
Looking at the case n = 5, there are at least 7 distinct potential quadrilateral arrangements—i.e., 7 different eligible cycle products of the numerals 0 through 4. I’m looking into the question of whether each of these represents a physically possible 3-d steepled quadrilateral arrangement, and if so, can it be accomplished with all convex quadrilaterals or not. (That is, are you forced to use any non-convex quadrilaterals to construct the arrangement.)
I’m planning on posting this as a question to Math StackExchange, but I prefer to first be confident I can answer the question asked within a month—due to the logistics of that site, it’s possible for a question to disappear after a month if no-one has answered it.