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Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the oriented triangle $\,\ z_1 \to z_2 \to z_3 \to z_1\,\ $ is non-degenerate and is oriented counterclockwise.

If we write $z_k$ as $x_k + i y_k$ ($k = 1,2,3$), then we can write this relation as $x_1 y_2 + x_2 y_3 + x_3 y_1 > x_2 y_1 + x_3 y_2 + x_1 y_3$.

But I am wondering whether, instead of defining the ternary relation on $\bf{C}$ in terms of the binary relation on $\bf{R}$, one can give a kind of axiomatic characterization of the ternary relation directly, via such properties as the following:

(1) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(w+z_1,w+z_2,w+z_3)$.

(2) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(wz_1,wz_2,wz_3)$, as long as $w \neq 0$.

Of course, to get such an axiomatization off the ground, one would need a symmetry-breaking axiom saying that $(0,1,i)$ is a counterclockwise cycle, or something of that kind.

This seems like such a natural project that I doubt I am the first to consider it. What's known about this line of investigation?

A similar question concerns the ternary relation on $\bf{C}$ defined by the relation $x_1 y_2 + x_2 y_3 + x_3 y_1 \geq x_2 y_1 + x_3 y_2 + x_1 y_3$. (Note that for this relation, the stricture $w \neq 0$ in property (2) is unnecessary.) I can't decide which of the two ternary relations is more natural, so if people have worked out axiomatic approaches to this relation on $\bf{C}$, I'd be interested in this as well.

Please feel free to add good tags for this problem. The "complex-geometry" tag isn't right, and neither is "soft-question" or "foundations". I've opted for "axioms", but I suspect the choice is sub-optimal.

Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the oriented triangle

$$ z_1 \to z_2 \to z_3 \to z_1 $$

is non-degenerate and is oriented counterclockwise.

If we write $z_k$ as $x_k + i y_k$ ($k = 1,2,3$), then we can write this relation as $x_1 y_2 + x_2 y_3 + x_3 y_1 > x_2 y_1 + x_3 y_2 + x_1 y_3$.

But I am wondering whether, instead of defining the ternary relation on $\bf{C}$ in terms of the binary relation on $\bf{R}$, one can give a kind of axiomatic characterization of the ternary relation directly, via such properties as the following:

(1) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(w+z_1,w+z_2,w+z_3)$.

(2) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(wz_1,wz_2,wz_3)$, as long as $w \neq 0$.

Of course, to get such an axiomatization off the ground, one would need a symmetry-breaking axiom saying that $(0,1,i)$ is a counterclockwise cycle, or something of that kind.

This seems like such a natural project that I doubt I am the first to consider it. What's known about this line of investigation?

A similar question concerns the ternary relation on $\bf{C}$ defined by the relation $x_1 y_2 + x_2 y_3 + x_3 y_1 \geq x_2 y_1 + x_3 y_2 + x_1 y_3$. (Note that for this relation, the stricture $w \neq 0$ in property (2) is unnecessary.) I can't decide which of the two ternary relations is more natural, so if people have worked out axiomatic approaches to this relation on $\bf{C}$, I'd be interested in this as well.

Please feel free to add good tags for this problem. The "complex-geometry" tag isn't right, and neither is "soft-question" or "foundations". I've opted for "axioms", but I suspect the choice is sub-optimal.

Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the oriented triangle from $z_1$ to $z_2$ to $z_3$ to $z_1$ is non-degenerate and is oriented counterclockwise.

If we write $z_k$ as $x_k + i y_k$ ($k = 1,2,3$), then we can write this relation as $x_1 y_2 + x_2 y_3 + x_3 y_1 > x_2 y_1 + x_3 y_2 + x_1 y_3$.

But I am wondering whether, instead of defining the ternary relation on $\bf{C}$ in terms of the binary relation on $\bf{R}$, one can give a kind of axiomatic characterization of the ternary relation directly, via such properties as the following:

(1) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(w+z_1,w+z_2,w+z_3)$.

(2) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(wz_1,wz_2,wz_3)$, as long as $w \neq 0$.

Of course, to get such an axiomatization off the ground, one would need a symmetry-breaking axiom saying that $(0,1,i)$ is a counterclockwise cycle, or something of that kind.

This seems like such a natural project that I doubt I am the first to consider it. What's known about this line of investigation?

A similar question concerns the ternary relation on $\bf{C}$ defined by the relation $x_1 y_2 + x_2 y_3 + x_3 y_1 \geq x_2 y_1 + x_3 y_2 + x_1 y_3$. (Note that for this relation, the stricture $w \neq 0$ in property (2) is unnecessary.) I can't decide which of the two ternary relations is more natural, so if people have worked out axiomatic approaches to this relation on $\bf{C}$, I'd be interested in this as well.

Please feel free to add good tags for this problem. The "complex-geometry" tag isn't right, and neither is "soft-question" or "foundations". I've opted for "axioms", but I suspect the choice is sub-optimal.

Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the oriented triangle $\,\ z_1 \to z_2 \to z_3 \to z_1\,\ $ is non-degenerate and is oriented counterclockwise.

If we write $z_k$ as $x_k + i y_k$ ($k = 1,2,3$), then we can write this relation as $x_1 y_2 + x_2 y_3 + x_3 y_1 > x_2 y_1 + x_3 y_2 + x_1 y_3$.

But I am wondering whether, instead of defining the ternary relation on $\bf{C}$ in terms of the binary relation on $\bf{R}$, one can give a kind of axiomatic characterization of the ternary relation directly, via such properties as the following:

(1) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(w+z_1,w+z_2,w+z_3)$.

(2) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(wz_1,wz_2,wz_3)$, as long as $w \neq 0$.

Of course, to get such an axiomatization off the ground, one would need a symmetry-breaking axiom saying that $(0,1,i)$ is a counterclockwise cycle, or something of that kind.

This seems like such a natural project that I doubt I am the first to consider it. What's known about this line of investigation?

A similar question concerns the ternary relation on $\bf{C}$ defined by the relation $x_1 y_2 + x_2 y_3 + x_3 y_1 \geq x_2 y_1 + x_3 y_2 + x_1 y_3$. (Note that for this relation, the stricture $w \neq 0$ in property (2) is unnecessary.) I can't decide which of the two ternary relations is more natural, so if people have worked out axiomatic approaches to this relation on $\bf{C}$, I'd be interested in this as well.

Please feel free to add good tags for this problem. The "complex-geometry" tag isn't right, and neither is "soft-question" or "foundations". I've opted for "axioms", but I suspect the choice is sub-optimal.

Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the oriented triangle from $z_1$ to $z_2$ to $z_3$ to $z_1$ is non-degenerate and is oriented counterclockwise.

If we write $z_k$ as $x_k + i y_k$ ($k = 1,2,3$), then we can write this relation as $x_1 y_2 + x_2 y_3 + x_3 y_1 > x_2 y_1 + x_3 y_2 + x_1 y_3$. But I am wondering whether, instead of defining the ternary relation on $\bf{C}$ in terms of the binary relation on $\bf{R}$, one can give a kind of axiomatic characterization of the ternary relation directly, via such properties as the following:

(1) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(w+z_1,w+z_2,w+z_3)$.

(2) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(wz_1,wz_2,wz_3)$, as long as $w \neq 0$.

Of course, to get such an axiomatization off the ground, one would need a symmetry-breaking axiom saying that $(0,1,i)$ is a counterclockwise cycle, or something of that kind.

This seems like such a natural project that I doubt I am the first to consider it. What's known about this line of investigation?

A similar question concerns the ternary relation on $\bf{C}$ defined by the relation $x_1 y_2 + x_2 y_3 + x_3 y_1 \geq x_2 y_1 + x_3 y_2 + x_1 y_3$. (Note that for this relation, the stricture $w \neq 0$ in property (2) is unnecessary.) I can't decide which of the two ternary relations is more natural, so if people have worked out axiomatic approaches to this relation on $\bf{C}$, I'd be interested in this as well.

Please feel free to add good tags for this problem. The "complex-geometry" tag isn't right, and neither is "soft-question" or "foundations". I've opted for "axioms", but I suspect the choice is sub-optimal.

Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the oriented triangle from $z_1$ to $z_2$ to $z_3$ to $z_1$ is non-degenerate and is oriented counterclockwise.

If we write $z_k$ as $x_k + i y_k$ ($k = 1,2,3$), then we can write this relation as $x_1 y_2 + x_2 y_3 + x_3 y_1 > x_2 y_1 + x_3 y_2 + x_1 y_3$.

But I am wondering whether, instead of defining the ternary relation on $\bf{C}$ in terms of the binary relation on $\bf{R}$, one can give a kind of axiomatic characterization of the ternary relation directly, via such properties as the following:

(1) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(w+z_1,w+z_2,w+z_3)$.

(2) If $(z_1,z_2,z_3)$ is a counterclockwise cycle, then so is $(wz_1,wz_2,wz_3)$, as long as $w \neq 0$.

Of course, to get such an axiomatization off the ground, one would need a symmetry-breaking axiom saying that $(0,1,i)$ is a counterclockwise cycle, or something of that kind.

This seems like such a natural project that I doubt I am the first to consider it. What's known about this line of investigation?

A similar question concerns the ternary relation on $\bf{C}$ defined by the relation $x_1 y_2 + x_2 y_3 + x_3 y_1 \geq x_2 y_1 + x_3 y_2 + x_1 y_3$. (Note that for this relation, the stricture $w \neq 0$ in property (2) is unnecessary.) I can't decide which of the two ternary relations is more natural, so if people have worked out axiomatic approaches to this relation on $\bf{C}$, I'd be interested in this as well.

Please feel free to add good tags for this problem. The "complex-geometry" tag isn't right, and neither is "soft-question" or "foundations". I've opted for "axioms", but I suspect the choice is sub-optimal.