Exploring the 4th Dimension: A Discussion

[Jaami M.]

Hello,
I have recently been very interested in learning about the 4th dimension (well the little we know about it) I have listened and discussed in some conversations about the topic and have a few thoughts and it questions.

-From my understanding the Tesseract is a 3rd dimensional Shadow depiction of a 4th dimensional cube.

-Also just how a series of squares make up a Cube, and a series of cubes make up a Tesseract.

-Here is more of what I am thinking and please help correct me or elaborate on this, but 4th dimensional creatures or objects emit 3 dimensional shadows...

Ex: if there was a 4D apple, the shadow of that 4D object will create a 3D apple. (what we see in our world)
is this valid because our 3D selves cast a 2D shadow... On a 2d plane.
-is this valid?-Also I know many people say that the 4th dimension is just time... But I am referring to a 4th Dimensional Spacial realm. If it is time that makes up the 4D please Elaborate..

Thank you.

 

[Bandersnatch]

In the context of Special and General Relativity, the fourth dimension is indeed time. This coupling of time and space is what produces certain effects, such as gravity.
Nothing stops you from thinking of only spatial dimensions, of course, as is often done when considering space-time is unnecessary.
With a fourth spatial dimension, we'd simply have a 5-dimensional space-time.

Regarding visualisations of 4D objects, rather than trying to answer here, let me direct you to this site:
http://eusebeia.dyndns.org/4d/vis/vis
It's got more answers than more people have questions. ;)

 

[Mark44]

With a fourth spatial dimension, we'd simply have a 5-dimensional space-time.

And in the context of mathematics, none of the dimensions has to be time. We can have vector spaces with an arbitrary number of spatial dimensions, none of which represents time.

 

[aikismos]

As all dimensions are mathematical counts, another interesting aspect of dimensionality in mathematics is the use of fractional dimensions.
(See https://en.wikipedia.org/wiki/Fractal_dimension).

 

[Mark44]

As all dimensions are mathematical counts

Do you mean "concepts"?

 

[aikismos]

Do you mean "concepts"?

According to Dictionary.com a dimension has the following definitions (emphasis mine):

"Mathematics. a property of space; extension in a given direction: A straight line has one dimension, a parallelogram has two dimensions, and a parallelepiped has three dimensions.1. the generalization of this property to spaces with curvilinear extension, as the surface of a sphere. 2. the generalization of this property to vector spaces and to Hilbert space. 3. the generalization of this property to fractals, which can have dimensions that are noninteger real numbers.4.extension in time.

"Topology. a magnitude that, independently or in conjunction with other such magnitudes, serves to define the location of an element within a given set, as of a point on a line, an object in a space, or an event in space-time.the number of elements in a finite basis of a given vector space."

"Physics. any of a set of basic kinds of quantity, as mass, length, and time, in terms of which all other kinds of quantity can be expressed; usually denoted by capital letters, with appropriate exponents, placed in brackets: "

To answer your question, a resounding "nope". Etymologically speaking, all highlighted words (whether measurable lengths of space or time, or Minkowski space-time, or magnitudes, vectors, number, or quantity) indicate countable phenomena. Dimensions are in essence that which we count or measure be it science or mathematics. The word literally means that arising out of measurement. Therefore any definition of dimension has to be connected to counting and number systems.

 

[Jaami M.]

Well are my thoughts on the visual depictions of the 4th dimension correct?

 

[aikismos]

Well are my thoughts on the visual depictions of the 4th dimension correct?

Well, by definition, you are talking about a metaphor when you are talking shadows and projections in the basic sense extended to constructing a tesseract as a 3-D object; after all shadows and projections as we understand them intuitively are exercises of visualization in three-dimensional space from a psychological perspective. The generalization of three dimensions to four dimensions, when dealing with a cube in four dimensions can be modeled in two distinct ways:

1) The first is a series of events through space-time (in which case a cube has vertices associated with concomitant instantaneous time values), this is how our brains conceptualize 4-D natively. Watching a cube move through space, we always see 8 vertices and remember them as the object moves. If we were to describe the motion of a cube through space over time, it might be in Minkowski space where all events are 4-tuples, i.e. (x,y,z,t).
2) The second is by a more complicated three-dimensional locus of points in which the number of vertices are increased in the same mathematical progression as from 1-D to 2-D to 3-D up to 4-D. These are also called projections in relation to each other, but are actually used as 3-D collections of points in order to understand them. For instance, the Schlegel diagram of the tesseract necessarily has 16 points at any instant where the tesseract can be folded into many of it's 3-D nets. Confusingly these nets are also a 3-D locus of identical number of vertices, and despite the fact they may seem unique permutations of points in a space, they can be considered combinations in regards to the the eight cubes the tesseract can be folded into.

In this way, we are using 3-dimensional space to capture some of the relations that occur when look at mapping relations between 1-tuples to 2-tuples and 2-tuples to 3-tuples.

I guess to answer your question more broadly about whether your views on the nature of 4-dimensional objects are right, yes and no.

 

[Hornbein]

Hello,
I have recently been very interested in learning about the 4th dimension (well the little we know about it) I have listened and discussed in some conversations about the topic and have a few thoughts and it questions.

-From my understanding the Tesseract is a 3rd dimensional Shadow depiction of a 4th dimensional cube.

-Also just how a series of squares make up a Cube, and a series of cubes make up a Tesseract.

-Here is more of what I am thinking and please help correct me or elaborate on this, but 4th dimensional creatures or objects emit 3 dimensional shadows...

Ex: if there was a 4D apple, the shadow of that 4D object will create a 3D apple. (what we see in our world)
is this valid because our 3D selves cast a 2D shadow... On a 2d plane.
-is this valid?-Also I know many people say that the 4th dimension is just time... But I am referring to a 4th Dimensional Spacial realm. If it is time that makes up the 4D please Elaborate..

Thank you.

Recently scientists have begun to say that our space is 3+1 dimensional, since time is so different from the other three dimensions. You are talking about a 4+1 realm. (I've learned that it is always necessary to clarify this first.) So 4D here means the four dimensions in 4+1D space.

We can't create or draw truly 4D objects. There are several popular systems for depicting them as 2D pseudo-3D images. None is "right," its a matter of taste. The way I see it is that all of these methods are encodings of 4D objects. My advice is to understand the encodings very well, well enough that you can work backward to what is being encoded. This is hard to do. Indeed, try the more basic problem of figuring out what a 3D shape is by looking at its 2D shadow. It isn't easy.

Ex: if there was a 4D apple, the shadow of that 4D object will create a 3D apple. (what we see in our world)
.

Hmmm. If you are in a 4D world then all surfaces are 3D. A shadow is a surface, so it would be 3D. Of course, a shadow would be an absence of light -- so it wouldn't look much like an apple.

If the surface upon which the shadow is cast is flat, then that 3D shadow will also be flat.

What I do is consider 4D rotations, which are quite different from 3D rotations. That is, I always found those drawings of tesseracts and so forth baffling. It is better to visualize a rotating tesseract, I think. I don't like the way this is normally done, though. I think it would be better to encode the extra dimension as color.

 

[Mark44]

As all dimensions are mathematical counts...

Do you mean "concepts"?

According to Dictionary.com a dimension has the following definitions (emphasis mine):

"Mathematics. a property of space; extension in a given direction: A straight line has one dimension, a parallelogram has two dimensions, and a parallelepiped has three dimensions.1. the generalization of this property to spaces with curvilinear extension, as the surface of a sphere. 2. the generalization of this property to vector spaces and to Hilbert space. 3. the generalization of this property to fractals, which can have dimensions that are noninteger real numbers.4.extension in time.

"Topology. a magnitude that, independently or in conjunction with other such magnitudes, serves to define the location of an element within a given set, as of a point on a line, an object in a space, or an event in space-time.the number of elements in a finite basis of a given vector space."

"Physics. any of a set of basic kinds of quantity, as mass, length, and time, in terms of which all other kinds of quantity can be expressed; usually denoted by capital letters, with appropriate exponents, placed in brackets: "

Thank you for pointing out the obvious, but none of these definitions uses the word "count".

To answer your question, a resounding "nope". Etymologically speaking, all highlighted words (whether measurable lengths of space or time, or Minkowski space-time, or magnitudes, vectors, number, or quantity) indicate countable phenomena.
Dimensions are in essence that which we count or measure be it science or mathematics. The word literally means that arising out of measurement.

Yes, that much is obvious, but you are conflating "count" with "measure", and they are not the same. A recipe that calls for flour tells you what amount (a measure) to put in, but doesn't tell you to count the flour grains. A dimension is a "mathematical count" of what?

Therefore any definition of dimension has to be connected to counting and number systems.

"Mathematical measures" I would buy, but "mathematical counts" sounds off to my ear -- and I am a longtime native speaker of English.