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Aaron Meyerowitz
Aaron Meyerowitz
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I'm not sure if this does what you want, but the subgroup of order $3$ in the additive group $(\mathbb{Z}_{57},+)$ is $\{0,19,38\}.$

To expand on that, one construction for the plane $\mathbb{P}_{7}$ of order $7$ is to take as points the elements of $\mathbb{Z}_{57}$ with lines are   $$\ell_k=[k,k+1,k+3,k+13,k+32,k+36,k+43,k+52]$$ for $k \in \mathbb{Z}_{57}.$ Of course the lines are unordered sets but keeping the order might make the the cyclic collineation $\phi:x \mapsto x+1$ easier to follow.

The triangle $T=T_0=\{0,19,38\}$ (mentioned above) has $19$ cosets $T_i=\{i,i+19,i+38\}$.

The lines obtained by extending the sides are

$\ell_6=[6,\ \ 7\ ,9,\ \ {\large 19 },{\large 38},42,49,1]$

$\ell_{25}=[25,26,34,{\large 38 },{\large 0},4,11,20]$

$\ell_{44}=[44,45,47,{\large 0 },{\large 19},23,30,39]$

If "the" center of a triangle $\{P,Q,R\}$ in some plane (like $\mathbb{R}^2$ or $\mathbb{P}_{7}$) is "the" solution $C$ to $C+C+C=P+Q+R$ then we see that in $\mathbb{P}_{7}$ (as constructed here) a triangle has three centers. $C,C+19,C+38$ for a unique $0 \leq C \leq 18.$ These are just the vertices of $T_C.$

So the special triangles could be described those which are the triangle of centers for some triangle. OR as those triangles which coincide with their own triangle of centers.

I'm not sure if this does what you want, but the subgroup of order $3$ in the additive group $(\mathbb{Z}_{57},+)$ is $\{0,19,38\}.$

To expand on that, one construction for the plane $\mathbb{P}_{7}$ of order $7$ is to take as points the elements of $\mathbb{Z}_{57}$ with lines are $$\ell_k=[k,k+1,k+3,k+13,k+32,k+36,k+43,k+52]$$ for $k \in \mathbb{Z}_{57}.$ Of course the lines are unordered sets but keeping the order might make the the cyclic collineation $\phi:x \mapsto x+1$ easier to follow.

The triangle $T=T_0=\{0,19,38\}$ (mentioned above) has $19$ cosets $T_i=\{i,i+19,i+38\}$.

The lines obtained by extending the sides are

$\ell_6=[6,\ \ 7\ ,9,\ \ {\large 19 },{\large 38},42,49,1]$

$\ell_{25}=[25,26,34,{\large 38 },{\large 0},4,11,20]$

$\ell_{44}=[44,45,47,{\large 0 },{\large 19},23,30,39]$

If "the" center of a triangle $\{P,Q,R\}$ in some plane (like $\mathbb{R}^2$ or $\mathbb{P}_{7}$) is "the" solution $C$ to $C+C+C=P+Q+R$ then we see that in $\mathbb{P}_{7}$ (as constructed here) a triangle has three centers. $C,C+19,C+38$ for a unique $0 \leq C \leq 18.$ These are just the vertices of $T_C.$

So the special triangles could be described those which are the triangle of centers for some triangle. OR as those triangles which coincide with their own triangle of centers.

I'm not sure if this does what you want, but the subgroup of order $3$ in the additive group $(\mathbb{Z}_{57},+)$ is $\{0,19,38\}.$

To expand on that, one construction for the plane $\mathbb{P}_{7}$ of order $7$ is to take as points the elements of $\mathbb{Z}_{57}$ with lines   $$\ell_k=[k,k+1,k+3,k+13,k+32,k+36,k+43,k+52]$$ for $k \in \mathbb{Z}_{57}.$ Of course the lines are unordered sets but keeping the order might make the cyclic collineation $\phi:x \mapsto x+1$ easier to follow.

The triangle $T=T_0=\{0,19,38\}$ (mentioned above) has $19$ cosets $T_i=\{i,i+19,i+38\}$.

The lines obtained by extending the sides are

$\ell_6=[6,\ \ 7\ ,9,\ \ {\large 19 },{\large 38},42,49,1]$

$\ell_{25}=[25,26,34,{\large 38 },{\large 0},4,11,20]$

$\ell_{44}=[44,45,47,{\large 0 },{\large 19},23,30,39]$

If "the" center of a triangle $\{P,Q,R\}$ in some plane (like $\mathbb{R}^2$ or $\mathbb{P}_{7}$) is "the" solution $C$ to $C+C+C=P+Q+R$ then we see that in $\mathbb{P}_{7}$ (as constructed here) a triangle has three centers. $C,C+19,C+38$ for a unique $0 \leq C \leq 18.$ These are just the vertices of $T_C.$

So the special triangles could be described those which are the triangle of centers for some triangle. OR as those triangles which coincide with their own triangle of centers.

Source Link
Aaron Meyerowitz
Aaron Meyerowitz
  • 30.1k
  • 1
  • 48
  • 104

I'm not sure if this does what you want, but the subgroup of order $3$ in the additive group $(\mathbb{Z}_{57},+)$ is $\{0,19,38\}.$

To expand on that, one construction for the plane $\mathbb{P}_{7}$ of order $7$ is to take as points the elements of $\mathbb{Z}_{57}$ with lines are $$\ell_k=[k,k+1,k+3,k+13,k+32,k+36,k+43,k+52]$$ for $k \in \mathbb{Z}_{57}.$ Of course the lines are unordered sets but keeping the order might make the the cyclic collineation $\phi:x \mapsto x+1$ easier to follow.

The triangle $T=T_0=\{0,19,38\}$ (mentioned above) has $19$ cosets $T_i=\{i,i+19,i+38\}$.

The lines obtained by extending the sides are

$\ell_6=[6,\ \ 7\ ,9,\ \ {\large 19 },{\large 38},42,49,1]$

$\ell_{25}=[25,26,34,{\large 38 },{\large 0},4,11,20]$

$\ell_{44}=[44,45,47,{\large 0 },{\large 19},23,30,39]$

If "the" center of a triangle $\{P,Q,R\}$ in some plane (like $\mathbb{R}^2$ or $\mathbb{P}_{7}$) is "the" solution $C$ to $C+C+C=P+Q+R$ then we see that in $\mathbb{P}_{7}$ (as constructed here) a triangle has three centers. $C,C+19,C+38$ for a unique $0 \leq C \leq 18.$ These are just the vertices of $T_C.$

So the special triangles could be described those which are the triangle of centers for some triangle. OR as those triangles which coincide with their own triangle of centers.