B Equation of Circle: Boundary or Surface?

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The equation of a circle, x^2 + y^2 = 1, describes a boundary rather than a surface, as it represents a 1D curve. In contrast, a sphere, defined by the equation x^2 + y^2 + z^2 = 1, describes a 2D surface in 3D space. The discussion highlights the distinction between dimensions, noting that points are 0D, lines/curves are 1D, shapes are 2D, and solids are 3D objects. It also emphasizes that while boundaries can exist for curves and surfaces, the terminology can vary based on context. Overall, understanding these distinctions is crucial for mathematical clarity.
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Describing boundary or a surface?
Equation of circle: ##x ^2+y ^2=1##. Is this equation describing boundary or a surface?

Thanks.
 
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What do you think?
 
Can a 2D object be a surface?
 
PeroK said:
What do you think?

phinds said:
Can a 2D object be a surface?
For 2D objects, like circle, the equation describes boundary..?
For 3D objects, like sphere, the equation: x ^2 + y ^2 + z ^2 = 1: describes surface..?
 
pairofstrings said:
For 2D objects, like circle, the equation describes boundary..?
For 3D objects, like sphere, the equation: x ^2 + y ^2 + z ^2 = 1: describes surface..?
What's the definition of a boundary?

Can a surface be a boundary?

https://en.wikipedia.org/wiki/Surface_(mathematics)

Can a curve be a boundary?

By the way, a circle is a 1D object (curve) and a sphere is a 2D object (surface).
 
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PeroK said:
Can a surface be a boundary?
PeroK said:
Can a curve be a boundary?
I think that curve or a surface can have a boundary.
So, the boundary of a circle can be represented as ##x^2 + y^2=1##?

PeroK said:
By the way, a circle is a 1D object (curve) and a sphere is a 2D object (surface).
I thought Sphere is considered as a 3D object. I was incorrect. So now, out of curiosity, please may I know what a 3D object could be like?

Thanks.
 
pairofstrings said:
I think that curve or a surface can have a boundary.
So, the boundary of a circle can be represented as ##x^2 + y^2=1##?I thought Sphere is considered as a 3D object. I was incorrect. So now, out of curiosity, please may I know what a 3D object could be like?

Thanks.
A maths student is a 3d object!

Mathematically it's called a solid. A ball, sphere plus the interior, is a 3d object.
 
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Thanks.
I was looking into points, lines, curves, shapes, solids.
Lines and curves can be referred to as 1D objects ? There is a notion of boundary here?
Shapes, like Sphere, Cone, Cube, Cylinder can be considered as 2D objects? There is a notion of surface here? I can calculate Surface area here?
Solids are 3D objects. There is a notion of surface, volume here. I can calculate surface area, volume here?
Point is a zero dimensional object.
 
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I think things are getting rather confused here. Words like 'Sphere, Cone, Cube, Cylinder' have different meanings according to their context.

pairofstrings said:
Lines and curves can be referred to as 1D objects ?
Yes.

pairofstrings said:
There is a notion of boundary here?
A line segment is bounded by two points. The boundary of a 2D shape is a (closed) curve.

pairofstrings said:
Shapes, like Sphere, Cone, Cube, Cylinder can be considered as 2D objects? There is a notion of surface here? I can calculate Surface area here?
Strictly speaking the word 'sphere' refers to a 2D surface, but I don't think that is helpful here. We normally consider shapes like triangles, squares etc. as 2 dimensional.

The shape of a coin is called a 'disk' and this is also a 2D object. The boundary of this shape is a 'circle' which, because it is a curve only has 1 dimension but we would not normally refer to a circle as a 1 dimensional object.

pairofstrings said:
Solids are 3D objects. There is a notion of surface area, volume here. I can calculate surface area, volume here?
Yes. When we say sphere, cone, cube, cylinder or prism we are usually referring to a 3D object.

When we want to avoid confusion we say "the surface of a [sphere, cone, cube, cylinder or prism]" or "the interior of a [sphere, cone, cube, cylinder or prism]" to refer to the 2D and 3D objects respectively.

pairofstrings said:
Point is a zero dimensional object.
Yes, a point has no dimensions.
 
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So, first there is a point then line/curve then shapes then solids according to ascending order of dimensions?

What is the point, line/curve, shape, solid collectively called? Are they called mathematical objects? Each one of them is a mathematical object?
 
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  • #11
pairofstrings said:
So, first there is a point then line/curve then shapes then solids according to ascending order of dimensions?
I don't like the word "shapes" here, perhaps you could use "plane figures".
pairofstrings said:
What is the point, line/curve, shape, solid collectively called? Are they called mathematical objects? Each one of them is a mathematical object?
No, the term "mathematical object" does not have any generally accepted meaning, and there isn't really a collective noun for these geometrical concepts except perhaps each of them could be called a "figure".

If you want to learn about geometry then you should learn about things that are important; asking random questions about things that may not be important will not help you learn. Khan Academy has a suitable set of courses in basic geometry.
 
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