Fano Geometry. Doesn't the intersection of the circle and the line constitute a point?

Welcome to Free Math Help!

Since circles are curved everywhere and lines are curved nowhere, the largest region of intersection they can share at a time is a point. A line tangent to a circle will intersect it at only one point, whereas a line that moves through the interior of a circle will intersect it at two points.

My crystal ball suggests there is a debate of some sort that prompted this posting. Perhaps a riddle or school assignment is asking about the dots in the image, referring to them as "points"? Perhaps you've proposed a solution that regards the intersection of the line and circle as a "point", yet the problem's authority doesn't accept it as an answer? It's true that the cite of intersection between circle and line is a point, but don't confuse the exact words for the spirit of the problem being presented.
 
Darius Markus said:
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Markus, you did not follow directions. You did not tell us what you need to know or what you have done.
Mr. Bland, you clearly have no idea what Fano's seven-point projective geometry is about.
Here are Fano's five axioms:
P1. There exists at least one line.
P2. There exists three points on every line.
P3. Not all points on the same line.
P4. There is exactly one line on any two points.
P5. There is at least one one point on any two lines.
Markus, I will not try to guess what question for which you are asking help. So tell us.
 
Darius Markus said:
Doesn't the intersection of the circle and the line constitute a point?
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You appear to be doing things backward. The diagram here does not define the geometry; it is a model of the geometry, derived from the axioms. So you can't make conclusions based on what you see, just as you can't conclude from the game Battleship that ships are made of plastic.

Perhaps even more important, this model is not to be thought of as existing on a Euclidean plane. The seven dots represent the only points in the geometry! You can't make a deduction from what you see, as if each line in the picture consisted of infinitely many points. Each line actually consists only of those three points marked on it. You should think of the lines drawn as mere representations of relationships: dots connected by a drawn curve represent points that are on the line represented by that curve.

If you have seen any graph theory, that would be an even better way to think of this picture. In graph theory, edges can appear to cross, but only points marked as nodes count. You can think of the lines as being shifted out of the plane so that they don't intersect.

For a good explanation of how the model is made starting from the axioms, see here:
The answer to the second question touches on what I have said.
 
Mr. Bland said:
Welcome to Free Math Help!

Since circles are curved everywhere and lines are curved nowhere, the largest region of intersection they can share at a time is a point. A line tangent to a circle will intersect it at only one point, whereas a line that moves through the interior of a circle will intersect it at two points.

My crystal ball suggests there is a debate of some sort that prompted this posting. Perhaps a riddle or school assignment is asking about the dots in the image, referring to them as "points"? Perhaps you've proposed a solution that regards the intersection of the line and circle as a "point", yet the problem's authority doesn't accept it as an answer? It's true that the cite of intersection between circle and line is a point, but don't confuse the exact words for the spirit of the problem being presented.
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Thanks!
 
Dr.Peterson said:
You appear to be doing things backward. The diagram here does not define the geometry; it is a model of the geometry, derived from the axioms. So you can't make conclusions based on what you see, just as you can't conclude from the game Battleship that ships are made of plastic.

Perhaps even more important, this model is not to be thought of as existing on a Euclidean plane. The seven dots represent the only points in the geometry! You can't make a deduction from what you see, as if each line in the picture consisted of infinitely many points. Each line actually consists only of those three points marked on it. You should think of the lines drawn as mere representations of relationships: dots connected by a drawn curve represent points that are on the line represented by that curve.

If you have seen any graph theory, that would be an even better way to think of this picture. In graph theory, edges can appear to cross, but only points marked as nodes count. You can think of the lines as being shifted out of the plane so that they don't intersect.

For a good explanation of how the model is made starting from the axioms, see here:
The answer to the second question touches on what I have said.
Click to expand...
Thank you! We were trying to come up with a model for Fano Geometry from the axioms. We got to the circle part but we got confused as to why there are four points in one line (we were counting the intersection as a point) while the axiom says there's exactly three per line. Thanks for the answer! It cleared everything up. ?
 
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I am trying to clarify whether the intersection of the circle and the line in the model (highlighted by the arrow) constitutes a point. The question I posted is my actual question. Thanks, though. ?
 
Darius Markus said:
Thank you! We were trying to come up with a model for Fano Geometry from the axioms. We got to the circle part but we got confused as to why there are four points in one line (we were counting the intersection as a point) while the axiom says there's exactly three per line. Thanks for the answer! It cleared everything up. ?
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Great! So you really were working in the right direction, but got into trouble by sort of "looking backward" after you got there! It's a subtle point that probably isn't stated often enough.
 
Darius Markus said:
I am trying to clarify whether the intersection of the circle and the line in the model (highlighted by the arrow) constitutes a point.
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The point is that there are no circles in Fano's geometry.
Fano's geometry has no metric thus a circle is an impossible concept.
There are only points and lines. A line may look like what you may call a an ordinary arc of a circle.
All of that means is for you must give up any ordinary concepts of geometry: they are no more.
If you are to join this mathematical game then you must play by the rules (axioms) .
 
pka said:
The point is that there are no circles in Fano's geometry.
Fano's geometry has no metric thus a circle is an impossible concept.
There are only points and lines. A line may look like what you may call a an ordinary arc of a circle.
All of that means is for you must give up any ordinary concepts of geometry: they are no more.
If you are to join this mathematical game then you must play by the rules (axioms) .
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Thank you! Yep. We got past that hurdle. What got us confused is the intersection of the circle (as a line) and the other lines. Thanks for clearing it up. ?
 
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