# How many dimensions fit in a point?

Discussion in 'Physics & Math' started by loverbal, Apr 13, 2018.

1. ### loverbalRegistered Member

Messages:
18
How many dimensions fit in a point?

My understanding is that a point is zero dimensions , xyz and time not existing in a point

In a volume of points there xyz but still no time?

3. ### DaveC426913Valued Senior Member

Messages:
11,130
Dimensions do not fit in things. Things fit in dimensions.

A point has, by definition, zero dimensions.

You can certainly have a volume filled with points, without having time. Check out almost any geometry lesson.
But it won't be physical - in the real world.

5. ### James RJust this guy, you know?Staff Member

Messages:
32,324
A point has zero dimensions. That's the mathematical definition, anyway.

7. ### Write4UValued Senior Member

Messages:
10,325

Would a better question not be to ask what potential can a single dimensionless point offer?.

8. ### James RJust this guy, you know?Staff Member

Messages:
32,324
This thread is in the maths forum, not Philosophy.

9. ### loverbalRegistered Member

Messages:
18
What do you mean by this James?

10. ### loverbalRegistered Member

Messages:
18
What do you mean by it will not be physical in the real world?

Is a perfect vacuum not xyz without time?

Does a point have space-time?

Do several points of a volume have space-time?

https://en.wikipedia.org/wiki/Spacetime

11. ### James RJust this guy, you know?Staff Member

Messages:
32,324
I mean that the idea of a dimensionless point is an abstraction. It's a useful concept, but no mathematical points actually exist in the physical world. A "point" is an idea - that's all.

Take something like Euclidean geometry. The concept of a dimensionless point is very useful for deriving mathematical results, but in the real world there are no zero-dimensional points, or one-dimensional lines, or perfect triangles, and no perfect right angles.

Similarly, it i useful to describe events in space and time using an imaginary co-ordinate system with x, y and z spatial coordinates and a time coordinate, but this is a model. The map is not the territory. It is just something we use to understand the territory.

12. ### Q-reeusValued Senior Member

Messages:
3,172
It may seem 'common sense' that a mathematical point = zero spatial dimensions, but you need to specify the available dimensionality. That xyz zero dimensional point will generally sweep out a 1D worldline in 4D spacetime. Unless you go further and specify zero temporal extent also, in which case you have picked out a 4D point in 4D spacetime.

13. ### James RJust this guy, you know?Staff Member

Messages:
32,324
Following on from above...
A vacuum is an absence of matter. The word "vacuum" refers to something happening (or rather, not happening) in a physical space. x, y and z are mathematical coordinates, which can be useful to model physical space conceptually. But the coordinates are not the space - they are just a way to stick labels on the space.

Not really. A point is just geometrical. Maybe you're thinking of something more like a spacetime event, which can have both spatial and time coordinates assigned to it.

Volume is a spatial concept, and is also a mathematical abstraction just like a point. Maybe it's better to think of volume as a property that space has, rather than the other way around. Or perhaps a property that we impose on space as a convenient way to describe it.

14. ### James RJust this guy, you know?Staff Member

Messages:
32,324
I should also add that mathematical concepts like "volume" can be extended into as many dimensions as you like, mathematically. We can talk about the volume of a 10 dimensional sphere, for example. But that says nothing about whether physical space actually has 10 degrees of freedom that we can describe using coordinates in 10 dimensions.

People who work with maths and with large data often talk about degrees of variation in the data as "dimensions", and they can validly talk about "volumes" in a given data set. Physicists sometimes talk about phase space, in which each "point" represents not just where a particle is located in space and time, but also how fast it is moving (for example). By a natural extension, they can talk about volumes in phase space.

The point is, it's all about building a mental model using mathematical concepts. Maths represents things in the physical world, but you should never confuse the mathematical objects with the physical objects.

15. ### loverbalRegistered Member

Messages:
18
This is a little bit confusing to me.

A volume consists of points?

A volume of points has no space-time or time unless the volume of points are occupied by matter?

Matter gives a point dimensions of xyz and time?

16. ### loverbalRegistered Member

Messages:
18
How can you have a 4d point when the answers have told me a point has 0 dimensions?

A 4d point would be a ''dot''? not a point

17. ### Q-reeusValued Senior Member

Messages:
3,172
How? A 0 dimensional point still has location within whatever dimensional space or spacetime it is embedded. We are talking mathematical use not 'real physics' - are we not?
Sounds like hair-splitting over a definition. If by dot is implied finite extent, then that implies a small but finite extent. Otherwise, a 4D point is a perfectly legitimate mathematical object within 4D spacetime.

exchemist likes this.
18. ### loverbalRegistered Member

Messages:
18
If you are describing a point as zero dimensions, it is then obviously contradictory describing a 4 dimensional point. To me that is saying it is no longer a point from what as been originally described. Giving the zero point dimensions, turning the point into a volume, a volume cannot have zero dimensions , by stating 4d you are stating it is not a point and has volume .

I am talking mathematical and physical. A volume of space exists, points in that volume exists, we can apply maths to the volume.

19. ### Q-reeusValued Senior Member

Messages:
3,172
Read the first few paras here: https://en.wikipedia.org/wiki/Spacetime#Definitions
I would personally distinguish 'event' from 'point' in that an event implies physics whereas a point can be empty of physics and just be a coordinate location. However the two seem generally to be used interchangeably in relativity circles. Don't get all hung over a personal definition that may be non-standard.

20. ### loverbalRegistered Member

Messages:
18
I am aware of the definition of space-time and what it means. Does space-time have physicality is the real question?

If a point has zero dimensions , zero time and zero physicality, then a volume of geometrical points must also have no physicality or time?

21. ### Q-reeusValued Senior Member

Messages:
3,172
If you understand the definition of spacetime how can there be such a question? The mathematical scaffolding necessarily uses idealized entities such as zero dimensional event/point, upon which real physics can be precisely described. A map vs territory situation if you like.
Are we going down a rabbit hole where trying to make sense of 0 x infinity leads to artificial contradictions? There are better reasons to get a headache.

22. ### C CConsular Corps - "the backbone of diplomacy"Valued Senior Member

Messages:
1,926
If referring to something that's abstractly well-mannered or ideal, it shouldn't have any degrees of freedom within it. A verbal / symbolic description or formulaic expression could define it without dimensions. But expressing it pictorially / empirically would require dependence upon an environment outside itself, of it being embedded in at least a minimal surrounding flat surface, and occupying the "space" of at least one pixel or equivalent fundamental unit. An arguably specious visual perspective from completely inside a point (absolute, no relationships to anything else) would not indicate what it is to an intruding observer, as well the "size-less" blankness or homogeneity (seemingly required to represent / instantiate it at all) sporting the erroneous option of being conceived as extended rather than a simple geometric element.

~

23. ### exchemistValued Senior Member

Messages:
7,840
Theorist back again?