Return to Question

I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point?

Specifically, given the points: $P_{ij}=(x_{ij},y_{ij}), 1\leq i,j \leq 3$, such that the circles $C_i$ are determined by $P_{ij}$, is there a formula based on the points $P_{ij}$ to determine whether the circles $C_i$ all intersect at one common point? If it helps, for the problem in question one may take $P_{ij}=P_{ji}$.

What would be particularly pleasing is if the formula could be represented as a determinant of a matrix (or potentially a determinant like object for a tensor). This would be reminiscent of the determinant condition for four points to be concyclic.

Edit: Given that the general problem has in some sense been solved, I would like to focus on a particular set of coordinates. The specific matrix I want to work with is:

\begin{pmatrix}(0,-y)&(0,y)&(x,0) \\ (0,y)&(-c,d)&(c,d) \\ (x,0)&(c,d)&(\frac{c^2+d^2}x,0)\end{pmatrix}

I am hoping to arrive at a diophantine equation in c, d, x, and y of small degree that would allow a full characterization (or perhaps an infinite family) of rational dodecahedron.

Edit 2: So I have used Gro-Tsen's degree six equation, and plugged in the points from the dodecahedron matrix. The resulting equation is of degree 8, and after removing the trivial factors $cy(c^2+d^2-x^2)$, the final equation is: $cxy^2-d^2y^2-c^2y^2+dx^2y+d^3y+c^2dy-d^2x^2-c^2x^2+cd^2x+c^3x=0$. You can rewrite this more simply as $xy(cy+dx)+(c^2+d^2)(cx+dy-x^2-y^2)=0$. This degree 4 Diophantine equation is the condition for a rational dodecahedron.

One final thing I would like to know... Is Gro-Tsen's formula for the symmetric 3x3 set of coordinates actually a determinant of a 6x6 matrix? It seems likely, given that it is degree six and has 720 terms (of which exactly half are negative). If so, what is that matrix?

I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point?

Specifically, given the points: $P_{ij}=(x_{ij},y_{ij}), 1\leq i,j \leq 3$, such that the circles $C_i$ are determined by $P_{ij}$, is there a formula based on the points $P_{ij}$ to determine whether the circles $C_i$ all intersect at one common point? If it helps, for the problem in question one may take $P_{ij}=P_{ji}$.

What would be particularly pleasing is if the formula could be represented as a determinant of a matrix (or potentially a determinant like object for a tensor). This would be reminiscent of the determinant condition for four points to be concyclic.

Edit: Given that the general problem has in some sense been solved, I would like to focus on a particular set of coordinates. The specific matrix I want to work with is:

\begin{pmatrix}(0,-y)&(0,y)&(x,0) \\ (0,y)&(-c,d)&(c,d) \\ (x,0)&(c,d)&(\frac{c^2+d^2}x,0)\end{pmatrix}

I am hoping to arrive at a diophantine equation in c, d, x, and y of small degree that would allow a full characterization (or perhaps an infinite family) of rational dodecahedron.

Edit 2: So I have used Gro-Tsen's degree six equation, and plugged in the points from the dodecahedron matrix. The resulting equation is of degree 8, and after removing the trivial factors $cy(c^2+d^2-x^2)$, the final equation is: $cxy^2-d^2y^2-c^2y^2+dx^2y+d^3y+c^2dy-d^2x^2-c^2x^2+cd^2x+c^3x=0$. You can rewrite this more simply as $xy(cy+dx)+(c^2+d^2)(cx+dy-x^2-y^2)=0$. This degree 4 Diophantine equation is the condition for a rational dodecahedron.

For anyone interested, I have programmed in desmos what the stereographic projection of the rational dodecahedron actually looks like. The points (c,d), (x,0), and (0,y) (which have been changed to (f,0) and (0,g)) are all adjustable. The final equation is the condition for co-intersection.

One final thing I would like to know... Is Gro-Tsen's formula for the symmetric 3x3 set of coordinates actually a determinant of a 6x6 matrix? It seems likely, given that it is degree six and has 720 terms (of which exactly half are negative). If so, what is that matrix?

I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point?

Specifically, given the points: $P_{ij}=(x_{ij},y_{ij}), 1\leq i,j \leq 3$, such that the circles $C_i$ are determined by $P_{ij}$, is there a formula based on the points $P_{ij}$ to determine whether the circles $C_i$ all intersect at one common point? If it helps, for the problem in question one may take $P_{ij}=P_{ji}$.

What would be particularly pleasing is if the formula could be represented as a determinant of a matrix (or potentially a determinant like object for a tensor). This would be reminiscent of the determinant condition for four points to be concyclic.

Edit: Given that the general problem has in some sense been solved, I would like to focus on a particular set of coordinates. The specific matrix I want to work with is:

\begin{pmatrix}(0,-y)&(0,y)&(x,0) \\ (0,y)&(-c,d)&(c,d) \\ (x,0)&(c,d)&(\frac{c^2+d^2}x,0)\end{pmatrix}

I am hoping to arrive at a diophantine equation in c, d, x, and y of small degree that would allow a full characterization (or perhaps an infinite family) of rational dodecahedron.

I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point?

Specifically, given the points: $P_{ij}=(x_{ij},y_{ij}), 1\leq i,j \leq 3$, such that the circles $C_i$ are determined by $P_{ij}$, is there a formula based on the points $P_{ij}$ to determine whether the circles $C_i$ all intersect at one common point? If it helps, for the problem in question one may take $P_{ij}=P_{ji}$.

What would be particularly pleasing is if the formula could be represented as a determinant of a matrix (or potentially a determinant like object for a tensor). This would be reminiscent of the determinant condition for four points to be concyclic.

Edit: Given that the general problem has in some sense been solved, I would like to focus on a particular set of coordinates. The specific matrix I want to work with is:

\begin{pmatrix}(0,-y)&(0,y)&(x,0) \\ (0,y)&(-c,d)&(c,d) \\ (x,0)&(c,d)&(\frac{c^2+d^2}x,0)\end{pmatrix}

I am hoping to arrive at a diophantine equation in c, d, x, and y of small degree that would allow a full characterization (or perhaps an infinite family) of rational dodecahedron.

Edit 2: So I have used Gro-Tsen's degree six equation, and plugged in the points from the dodecahedron matrix. The resulting equation is of degree 8, and after removing the trivial factors $cy(c^2+d^2-x^2)$, the final equation is: $cxy^2-d^2y^2-c^2y^2+dx^2y+d^3y+c^2dy-d^2x^2-c^2x^2+cd^2x+c^3x=0$. You can rewrite this more simply as $xy(cy+dx)+(c^2+d^2)(cx+dy-x^2-y^2)=0$. This degree 4 Diophantine equation is the condition for a rational dodecahedron.

One final thing I would like to know... Is Gro-Tsen's formula for the symmetric 3x3 set of coordinates actually a determinant of a 6x6 matrix? It seems likely, given that it is degree six and has 720 terms (of which exactly half are negative). If so, what is that matrix?

I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point?

Specifically, given the points: $P_{ij}=(x_{ij},y_{ij}), 1\leq i,j \leq 3$, such that the circles $C_i$ are determined by $P_{ij}$, is there a formula based on the points $P_{ij}$ to determine whether the circles $C_i$ all intersect at one common point? If it helps, for the problem in question one may take $P_{ij}=P_{ji}$.

What would be particularly pleasing is if the formula could be represented as a determinant of a matrix (or potentially a determinant like object for a tensor). This would be reminiscent of the determinant condition for four points to be concyclic.

Edit: Given that the general problem has in some sense been solved, I would like to focus on a particular set of coordinates. The specific matrix I want to work with is:

\begin{pmatrix}(0,-y)&(0,y)&(x,0) \\ (0,y)&(-a,b)&(a,b) \\ (x,0)&(a,b)&(\frac{a^2+b^2}x,0)\end{pmatrix}

I am hoping to arrive at a diophantine equation in a, b, x, and y of small degree that would allow a full characterization (or perhaps an infinite family) of rational dodecahedron.

One sanity check that this actually works: a=22, b=54, x=40, y=10 should be a solution (because it does generate a rational dodecahedron). I tried working this out myself but I keep not getting it equal to zero.

I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point?

Specifically, given the points: $P_{ij}=(x_{ij},y_{ij}), 1\leq i,j \leq 3$, such that the circles $C_i$ are determined by $P_{ij}$, is there a formula based on the points $P_{ij}$ to determine whether the circles $C_i$ all intersect at one common point? If it helps, for the problem in question one may take $P_{ij}=P_{ji}$.

What would be particularly pleasing is if the formula could be represented as a determinant of a matrix (or potentially a determinant like object for a tensor). This would be reminiscent of the determinant condition for four points to be concyclic.

Edit: Given that the general problem has in some sense been solved, I would like to focus on a particular set of coordinates. The specific matrix I want to work with is:

\begin{pmatrix}(0,-y)&(0,y)&(x,0) \\ (0,y)&(-c,d)&(c,d) \\ (x,0)&(c,d)&(\frac{c^2+d^2}x,0)\end{pmatrix}

I am hoping to arrive at a diophantine equation in c, d, x, and y of small degree that would allow a full characterization (or perhaps an infinite family) of rational dodecahedron.