The Next Step (Plane Geometry) After Defining the Family of Distance Measuring Rays in the Cartesian Plane?
We start with $(M,0,+) left( ; = (mathbb R^{ge 0},0,+) ; right)$, a system of magnitudes with an additive identity.
Consider linear isometric mappings $rho: M to mathbb R^2$, where the image of the mapping is a ray and the distance between any points $m.n in M$ is equal to the length of the line segment
joining $rho(m)$ and $rho(n)$.
Intuitively (to me anyway), the collection (and relations between rays and points) of all these isometric-rays describe the plane geometry. So the hunch is that we can abstract this and replace $mathbb R^2$ with a set of points $mathbb E^2 = text{Euclid's 2-Dim Plane}$ and a family of 'isometric-rays' satisfying a list of properties/axioms. In this system, the unit of measure on the plane is not defined but can be fixed when convenient. This makes sense - similar triangle can be defined using proportions, and a selected unit of measure is not required.
Has such a program been carried out?
I found this
Old and New Results in the
Foundations of
Elementary Plane Euclidean and
Non-Euclidean Geometries
by Marvin Jay Greenberg
but it was a recondite article and I didn't to see (on a quick review) any immediate connections.
For more along the same ideas, see the accepted answer from stack question
The status of high school geometry
soft-question axioms plane-geometry
asked yesterday
CopyPasteIt
4,0501627
add a comment |
We start with $(M,0,+) left( ; = (mathbb R^{ge 0},0,+) ; right)$, a system of magnitudes with an additive identity.
Consider linear isometric mappings $rho: M to mathbb R^2$, where the image of the mapping is a ray and the distance between any points $m.n in M$ is equal to the length of the line segment
joining $rho(m)$ and $rho(n)$.
Intuitively (to me anyway), the collection (and relations between rays and points) of all these isometric-rays describe the plane geometry. So the hunch is that we can abstract this and replace $mathbb R^2$ with a set of points $mathbb E^2 = text{Euclid's 2-Dim Plane}$ and a family of 'isometric-rays' satisfying a list of properties/axioms. In this system, the unit of measure on the plane is not defined but can be fixed when convenient. This makes sense - similar triangle can be defined using proportions, and a selected unit of measure is not required.
Has such a program been carried out?
I found this
Old and New Results in the
Foundations of
Elementary Plane Euclidean and
Non-Euclidean Geometries
by Marvin Jay Greenberg
but it was a recondite article and I didn't to see (on a quick review) any immediate connections.
For more along the same ideas, see the accepted answer from stack question
The status of high school geometry
soft-question axioms plane-geometry
asked yesterday
CopyPasteIt
4,0501627
add a comment |
We start with $(M,0,+) left( ; = (mathbb R^{ge 0},0,+) ; right)$, a system of magnitudes with an additive identity.
Consider linear isometric mappings $rho: M to mathbb R^2$, where the image of the mapping is a ray and the distance between any points $m.n in M$ is equal to the length of the line segment
joining $rho(m)$ and $rho(n)$.
Intuitively (to me anyway), the collection (and relations between rays and points) of all these isometric-rays describe the plane geometry. So the hunch is that we can abstract this and replace $mathbb R^2$ with a set of points $mathbb E^2 = text{Euclid's 2-Dim Plane}$ and a family of 'isometric-rays' satisfying a list of properties/axioms. In this system, the unit of measure on the plane is not defined but can be fixed when convenient. This makes sense - similar triangle can be defined using proportions, and a selected unit of measure is not required.
Has such a program been carried out?
I found this
Old and New Results in the
Foundations of
Elementary Plane Euclidean and
Non-Euclidean Geometries
by Marvin Jay Greenberg
but it was a recondite article and I didn't to see (on a quick review) any immediate connections.
For more along the same ideas, see the accepted answer from stack question
The status of high school geometry
soft-question axioms plane-geometry
asked yesterday
CopyPasteIt
4,0501627
We start with $(M,0,+) left( ; = (mathbb R^{ge 0},0,+) ; right)$, a system of magnitudes with an additive identity.
Consider linear isometric mappings $rho: M to mathbb R^2$, where the image of the mapping is a ray and the distance between any points $m.n in M$ is equal to the length of the line segment
joining $rho(m)$ and $rho(n)$.
Intuitively (to me anyway), the collection (and relations between rays and points) of all these isometric-rays describe the plane geometry. So the hunch is that we can abstract this and replace $mathbb R^2$ with a set of points $mathbb E^2 = text{Euclid's 2-Dim Plane}$ and a family of 'isometric-rays' satisfying a list of properties/axioms. In this system, the unit of measure on the plane is not defined but can be fixed when convenient. This makes sense - similar triangle can be defined using proportions, and a selected unit of measure is not required.
Has such a program been carried out?
I found this
Old and New Results in the
Foundations of
Elementary Plane Euclidean and
Non-Euclidean Geometries
by Marvin Jay Greenberg
but it was a recondite article and I didn't to see (on a quick review) any immediate connections.
For more along the same ideas, see the accepted answer from stack question
The status of high school geometry
soft-question axioms plane-geometry
soft-question axioms plane-geometry
asked yesterday
CopyPasteIt
4,0501627
asked yesterday
CopyPasteIt
4,0501627
asked yesterday
CopyPasteIt
4,0501627
asked yesterday
CopyPasteIt
4,0501627
asked yesterday
CopyPasteIt
4,0501627
4,0501627
add a comment |
add a comment |
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