What is the definition of a plane?

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I've been thinking about the flat surface we call a plane. I've looked for definitions and none of the ones I have found satisfy me...in this respect...they seem dependent on assumed meanings and not self-dependent using only geometric contructions. Let me explain.

One definition of a plane says a non-curved surface that includes three non-co-linear points. Ok fine. But how can we define a non-curved surface using simple constructions without relying on the assumed idea of non-curved.

It's one thing to say a non-curved surface that includes 3 non-co-linear points but how can the 'flatness' of the surface we refer to as non-curved be defined using only geometric constructions? What makes flatness using only geometric terms?

Here's my try...a surface that includes 3 non-co-linear points, A, B, and C, and also includes all of the points contained by the three lines a, b, and c, passing through each pair of points A, B, and C, as well as including all of the points contained by all of the lines passing through any point on one of those three lines a, b, or c, and any other point on either of the other two original lines.

This seems to lay out three points...create 3 lines and an infinity of other lines all in a 2 dimensional plane. There are only one problem, how can we define the 'straightness' of a line (which is necessary to assume in my definition) by using only geometric constructions and that doesn't require the use of the term 'one dimensional) or assume an axiomatic concept of 'straightness'? This even requires the definition of linear. How do you define linear using only geometric construction and how can we define a plane?

In defining a sphere, for example, we can say that it is an object made up of only those points equidistant from a reference point. This is a self-contained definition that relies only on geometric construction. How do we define a line or plane in the same way?

tex

RaulTheUCSCSlug
Gold Member
I think you are thinking of it in reverse. You need things like linear algebra, planes, lines, etc. to define geometric shapes, not the other way around.

Have you taken a multivariable course? Or have you heard of the idea of domains and sets of values living in a "neighborhood" of another domain?

No I haven't taken any multivariable course yet. My definition of a sphere did not require that. Give me a point and a ruler and I will create any size sphere you like. The reason is that the properties are producible by geometric construction. My question is simply this...is it possible to define and prove the flatness of a plane or the linearity of a line by what I refer to as a geometric construction. In my example above I believe my definition of a plane does this...EXCEPT...that it requires the assumption that a line is 'straight' or non-bending. I can say that a line goes through 2 points but what makes it forever 'straight' unless we use some concept or axiom that says 'a lines has no curves'.

But how can I construct a line without the use of any math proofs by geometric construction which results in an object that IS IN FACT geometrically straight? I can say it goes through 2 points...but then what? What in the definition requires by construction that the line must continue straight-ly beyond those two points. I could say that a line is defined by all points equal distant from the x axis but then I would need a coordinate system and a plane on which those points reside. How can I say "a line is defined by all of the points that......" fill in the blank. I've seen definitions of a line that say that it's the intersection of two planes. What defines a plane which definition is self-dependent like a sphere? Going back to the sphere...it doesn't need any other thing to define the object...it can be constructed geometrically with no math required.

Of course, mathematically, I can define a line a y=mx+b. Yes. Any line I draw on a sheet of paper can be defined this way but it is in relation to an x/y coordinate system. If I take a ruler and draw a line on that paper by tracing through two dots on the page I draw a line....keeping in mind I am depending on the straightness of the ruler.

I think the definition of a plane depends, in my simple thinking, on the definition of a line. I can define sphericallity by simple geometry. Can the same be done with straightness or flatness without relying on given axioms?

I am beginning to believe that it is not possible. So just in case I'm calling this "The Tex Conjecture". Which states that other than point and a sphere there are no other geometric shapes which can be defined by a simple geometric construction and that does not rely on some given axiomatic rule. I'm kidding. I really would like to know if it's possible.

tex

RaulTheUCSCSlug
Gold Member
When you say sphere, do you mean a 3D sphere, or a 2D circle that lives on a plane. You can define the sphere itself as a plane, and in calc 3 you learn that you can define shapes and do "pull backs" or "push forwards" in which you can stretch an object/shape/line and make it one to one to a function. I would argue that a sphere cannot be defined without the use of planes and 3D space.

A sphere can be defined to be mapped onto a line. In that sense a line can be defined as a set of points in which can be mapped on to a geometric shape.

I might be lost in your thought process since it seems that lines are very much required in defining things geometrically. A line does not need to be straight, but a straight line is always made between two points if the line is the shortest distance between those two points. This always works in 1D, 2D, nd 3D. You cannot find an example where it does not work.

Two lines intersecting will also always form a plane. I think you might need to look into a multivariable course for clarification on these type of things.

I mean a 3d sphere. Your definition that a line is the shortest distance between two points would qualify as a non math definition. Better stated might be...a line consists of all of the points along the shortest path between two points. That's easy to understand. But what makes the line extend straight-ly beyond those two points that can be verbally described in the same fashion and by some similar definition so that we describe a line that extends straight-ly in both directions to infinity beyond those two points? If we need points to define a line and a line or more to define a plane then we have to have some easily quotable definition for a line that goes from infinity to infinity and by the very definition (much like the shortest distance rule) defines and guarantees a line's straightness from infinity to infinity.

Two lines intersecting do not form a plane unless you accept the axiom that the resulting object is accepted as flat across the entire surface. But how do we create that flat plane, by a definition, that extends, by construction, from infinity to infinity? Some parts of the plane laying across those two lines might be flat but farther out the plane begins to curve...unless you can define a geometric construction that defines all of the points on a plane as those points.........(fill in the blank) similar to the definition of a sphere or simply accept by axiom that it is flat.

With the sphere definition we do not need to rely on any axiom or pre-accepted concept to create a sphere. You simply put a point everywhere you can equidistantly from a reference point and, voila, you have a sphere, perfectly, and mathematically spherical. And I didn't have to graduate from 3rd grade arithmetic to do it or understand it.

Can that be done with a line or a plane, or any other geometric object for that matter?

I just thought of one.....

Given any two points in space two spheres may be created of certain radius about those points. Assuming the two spheres are of equal radius (which can be guaranteed by geometric construction) and are of sufficient radius to overlap, the two spheres will create a circle at their junction. A third sphere centered on a third point equally distant from the other two points, if of equal size to the other two spheres, will cut the original circle at two points. By expanding the size of the three spheres indefinitely and proportionately while maintaining the position of the three reference points we can define a line as all of the points created by the intersection of the three spheres at their common two points. This, I think, gives us all of the points along an infinite line and guarantees it's straight-ness. See what I mean?

Now can something similar be done with a plane?

tex

I guess I just answered my own question. Two spheres create a circle that can just as easily be extended indefinately. That circular plane can be defined as all of the lines connecting any two points around the circumference of the circle. And that can be taken to infinity.

so...

tex

Mark44
Mentor
When you say sphere, do you mean a 3D sphere, or a 2D circle that lives on a plane.
The usual definition of sphere is that it is a three-dimensional object all of whose points are equidistant from its center. The term "sphere" is sometimes used in mathematical contexts of a one-dimensional space (two points on a line equidistant from the center), two-dimensional space (a circle), or in spaces of three or more dimensions.
RaulTheUCSCSlug said:
You can define the sphere itself as a plane
???
RaulTheUCSCSlug said:
, and in calc 3 you learn that you can define shapes and do "pull backs" or "push forwards" in which you can stretch an object/shape/line and make it one to one to a function. I would argue that a sphere cannot be defined without the use of planes and 3D space.

A sphere can be defined to be mapped onto a line. In that sense a line can be defined as a set of points in which can be mapped on to a geometric shape.

I might be lost in your thought process since it seems that lines are very much required in defining things geometrically. A line does not need to be straight
To my way of thinking, "line" and "straight line" are synonomous, and calling it "straight" is redundant. A one-dim. shape that isn't straight is a curve.
RaulTheUCSCSlug said:
, but a straight line is always made between two points if the line is the shortest distance between those two points.
That's not a line -- it's a line seqment. A line is infinitely long in both directions.
RaulTheUCSCSlug said:
This always works in 1D, 2D, nd 3D. You cannot find an example where it does not work.

Two lines intersecting will also always form a plane. I think you might need to look into a multivariable course for clarification on these type of things.
Or linear algebra. A line is the figure generated by all scalar multiples of a particular vector. A plane is the figure generated by the sum of scalar multiples of two linearly independent vectors.

RaulTheUCSCSlug
Wouldn't the vector definition require a vector which requires a coordinate system as reference which would not be a geometric construction. But thank you, I didn't know that. I understand that the definition of a line assumes a straight line but that assumption is not a geometric construction. It is given that we are talking about a straight line only because we make that axiomatic. In other words I don't have to rely on any math to know that all of the points equal distant from a point creates a perfect sphere. What is a description of a process that constructs an object of infinite length where all of the points that make up that object are co-linear. There I go again...now we need to define co-linear by construction just like the term spherical by construction.

I want to create a line by the use of geometric construction similar to constructing a bisector that can be made with arcs centered on two points on a line segment. It doesn't require math. A line is infinitely long, yes, but its straightness is a given because we say it's straight. How can I draw a line with a verbal description of a geometric construction like I can with a sphere is the question? I think I came up with one way in my last post. Maybe I'm not expressing myself well.

tex

Mark44
Mentor
Wouldn't the vector definition require a vector which requires a coordinate system as reference
Not necessarily. You could have a vector in some arbitrary space, with no coordinates assumed. As sort of an example of what I'm talkiing about, there are two ways to calculate the dot product of two vectors: 1) a coordinate version, where if ##\vec{u} = <u_1, u_2, u_3>## and ##\vec{v} = <v_1, v_2, v_3>##, then ##\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3##; 2) a coordinate-free version, where ##\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos(\theta)## where ##\theta## is the angle between the two vectors.
thetexan said:
which would not be a geometric construction. But thank you, I didn't know that. I understand that the definition of a line assumes a straight line but that assumption is not a geometric construction. It is given that we are talking about a straight line only because we make that axiomatic.
Or by definition.
thetexan said:
In other words I don't have to rely on any math to know that all of the points equal distant from a point creates a perfect sphere. What is a description of a process that constructs an object of infinite length where all of the points that make up that object are co-linear. There I go again...now we need to define co-linear by construction just like the term spherical by construction.
I don't understand why you are trying to do this. Are you trying to come up with a process that would describe each point on a line?
thetexan said:
I want to create a line by the use of geometric construction similar to constructing a bisector that can be made with arcs centered on two points on a line segment. It doesn't require math. A line is infinitely long, yes, but its straightness is a given because we say it's straight.
And by how a line is defined.
thetexan said:
How can I draw a line with a verbal description of a geometric construction like I can with a sphere is the question? I think I came up with one way in my last post. Maybe I'm not expressing myself well.
I'm not understanding the emphasis on geometric construction. The Greeks pioneered geometry around 500 BC or so, but analytic geometry combines algebra with the concepts of geometry. Not very much math these days relies totally on principles of geometry alone.

RaulTheUCSCSlug
It began as me trying to define a plane. I knew it involved a flat surface (this is where the first problem hit. What defines flat?) that includes 3 non-co-linear points. It occurred to me that I could image a surface much like a table clothe draped over a table that might include the three points but that doesn't make it flat infinitely since the surface could include the 3 points and be curved. So I asked myself what can I say to define a surface that makes it flat in the definition. It would say something like this.....'a plane consists of all of the points that.......' and I would fill in the blank. Just like the sphere example. A sphere is all of the points equal distant from a reference point. But the harder I tried to come up with a verbal definition the harder it seemed without using axioms or givens in the definition itself. In my definition a few posts above I think I came up with a way to do it but then I realized even that definition requires a similarly constructive definition of a line. So a line became the problem.

A line segment can be defined this way by stating that all of the points laying along the shortest path between two points create a line segment and defines straightness but then how do I extend that beyond those two points without becoming axiomatic? I can define a line segment of 2 miles in length this way and by that definition we also define straightness between those two points. But if I use those two points 2 miles apart how do I define an infinite line going through those 2 points that is straight beyond those two points without saying something that uses the term 'straight'. If I stipulate that I will use all of the points laying along the shortest path between those two points (which thus defines linearity) by what definition do I extend that linearity beyond those two points? I could pick any two points infinitely far apart, I suppose, and thus define a set of co-linear points along the shortest path between those two infinitely separated points.

If I have three billiard balls on a table how do I define a way to ensure the three balls are touching and co-linear? We'll accept the surface of the table as solving one of the dimensions and concentrate on the problem in two dimensions. What about this...'set the three balls together such that the third ball is no closer to the first ball on one side than it is on the other.'? I could say 'the three balls are co-linear when they form only one silhouette' but this wouldn't work with points. So how can I describe a process that puts the balls into co-linearity by using nothing but geometric construction processes. And how do I define the set of points that create a line by only relying on the construction process?

But there's always the problem of how to define linearity beyond the two points which doesn't rely on the linearity between the two points. This kind of axiomatic assumption is everywhere in geometry definitions. A ray is a line extending infinitely from a single point. Even this simple wording presumes a 'straight' line. But how can we describe the construction of a straight line of infinite length? I believe I fell into the answer above by using simple geometric objects and construction which would produce the set of all points which are co-linear (and the process even defines co-linearity) and thus a line of infinite length and straightness.

Once the line problem is solved then the line can be used in a definition to construct a plane. All of this without the use of math and coordinate systems.

That is what got me started on this. Then it occurs to me how many other geometric shapes of either 2d or 3d can be defined this way. Or can they. How does one construct a tetrahedron using only geometric construction which can be verbally and simply defined or does it require the use of mathematical tools and coordinate systems to do so?

tex

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Mark44
Mentor
It began as me trying to define a plane.
Definitions come to mind. Here's one from wikipedia: A plane is an abstract surface which has infinite width and length, zero thickness, and zero curvature.
thetexan said:
I knew it involved a flat surface (this is where the first problem hit. What defines flat?)
A curvature of zero.
thetexan said:
that includes 3 non-co-linear points. It occurred to me that I could image a surface much like a table clothe draped over a table that might include the three points but that doesn't make it flat infinitely since the surface could include the 3 points and be curved. So I asked myself what can I say to define a surface that makes it flat in the definition. It would say something like this.....'a plane consists of all of the points that.......' and I would fill in the blank. Just like the sphere example. A sphere is all of the points equal distant from a reference point. But the harder I tried to come up with a verbal definition the harder it seemed without using axioms or givens in the definition itself. In my definition a few posts above I think I came up with a way to do it but then I realized even that definition requires a similarly constructive definition of a line. So a line became the problem.

A line segment can be defined this way by stating that all of the points laying along the shortest path between two points create a line segment and defines straightness but then how do I extend that beyond those two points without becoming axiomatic? I can define a line segment of 2 miles in length this way and by that definition we also define straightness between those two points. But if I use those two points 2 miles apart how do I define an infinite line going through those 2 points that is straight beyond those two points without saying something that uses the term 'straight'. If I stipulate that I will use all of the points laying along the shortest path between those two points (which thus defines linearity) by what definition do I extend that linearity beyond those two points? I could pick any two points infinitely far apart, I suppose, and thus define a set of co-linear points along the shortest path between those two infinitely separated points.
Given two distinct points A and B, extend the line segment ##\bar{AB}## infinitely far in both directions. Voila, you have a line.
thetexan said:
If I have three billiard balls on a table how do I define a way to ensure the three balls are touching and co-linear?
The centers of the three balls all lie along the same line and the distance between any two adjacent balls is zero.
thetexan said:
We'll accept the surface of the table as solving one of the dimensions and concentrate on the problem in two dimensions. What about this...'set the three balls together such that the third ball is no closer to the first ball on one side than it is on the other.'? I could say 'the three balls are co-linear when they form only one silhouette' but this wouldn't work with points. So how can I describe a process that puts the balls into co-linearity by using nothing but geometric construction processes.
Why are geometric construction processes important?
thetexan said:
And how do I define the set of points that create a line by only relying on the construction process?

But there's always the problem of how to define linearity beyond the two points which doesn't rely on the linearity between the two points. This kind of axiomatic assumption is everywhere in geometry definitions. A ray is a line extending infinitely from a single point. Even this simple wording presumes a 'straight' line. But how can we describe the construction of a straight line of infinite length? I believe I fell into the answer above by using simple geometric objects and construction which would produce the set of all points which are co-linear (and the process even defines co-linearity) and thus a line of infinite length and straightness.

Once the line problem is solved then the line can be used in a definition to construct a plane. All of this without the use of math and coordinate systems.
But why?
thetexan said:
That is what got me started on this. Then it occurs to me how many other geometric shapes of either 2d or 3d can be defined this way. Or can they. How does one construct a tetrahedron using only geometric construction which can be verbally and simply defined or does it require the use of mathematical tools and coordinate systems to do so?

tex

Just because it's fun to figure these things out. Please don't take this so seriously. It's a mental exercise...a thing of interest.

Thanks for the help
Tex

Nidum
Gold Member
(1)

Three actual physical flat planes are each true planes if when tested against each other they make contact at all points .

This is the basis of how high accuracy master surface plates and straight edges are made and tested .

(2)

Is there a way to convert the practical definition above into a mathematical definition of a perfectly flat surface ?

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fresh_42
Mentor
Is there a way to convert the practical definition above into a mathematical definition ?
I very much doubt that. An example is always a bad idea for turning it into a definition. Mathematical (Euclidean) planes have no boundaries. You cannot test to infinity. So a flat pan which locally is a plane fails to be one further apart from its center. What to do in higher dimensions? How to define non-Euclidean geometries? And what is contact? Will convergence be allowed?

Even the physical statement of contact is sloppy. They do not have real contact for otherwise they would fell through each other. And allowing electrical forces to be neglected do two plates of a capacitor have contact? Probably not, for it won't be a capacitor anymore. So what is the difference between a non-contact in a capacitor and a contact between planes that are equally held apart by electromagnetic forces?

Samy_A