If a fourth spatial dimension exists

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Stories about Columbus gave a good demonstration how dimensions work, as Columbus didn't sail off the edge of the Earth.

I know we can't see water boil in a dark room, but it's a moderately safe bet that the water will boil around a 100 degrees C at standard air pressure even in a dark room.

Maybe basic Dimensional Analysis will help: "Most physical quantities can be expressed in terms of combinations of five basic dimensions. These are mass (M), length (L), time (T), electrical current (I), and temperature, represented by the Greek letter theta (q). " From: http://www.physics.uoguelph.ca/tutorials/dimanaly/

Per "If A Fourth Spatial Dimension Exists", confusion over the coordinate system must be avoided too, not just the number of independent vectors required to determine a unique "point". A simple example is the surface of a sphere requires two distinct coordinates to determine a point on the sphere, but three vectors are involved, and in a convenient coordinate system, one of the three vectors can be held constant and ignored, since the restriction is placed of the example of remaining on the surface of the sphere. Mapping the surface of the sphere to the "flat Euclidean plane" for a conventional map works good for a small area, but for larger areas, the distortions become great, as with a Mercator Projection Map or a Great Circle Map of the Earth.

A coordinate system does not require orthogonal vectors (orthogonal vectors tend to be the most convenient to use though), as only the minimal number of independent vectors are required. In the sphere example, on leaving the surface of the sphere, the vector conveniently held constant previously, must then be considered, and 3 dimensions are then required to determine a "point", say the Longitude, the Latitude, and the altitude for locating the point. Since the altitude is now regarded as to vary with the Longitude and Latitude, the sphere now has a different degree of curvature for areas around a particular Longitude and Latitude. On a well mapped/known surface, the curvature can be recorded as an attribute with two components of the fixed surface (Gauss Curvature, which has a maximum with orthogonal vectors in the tangent plane, as a simple standardized measurement of a vector product in two dimensions). The particular location on the surface of the Earth still only requires two numbers, but the "manifold" (as distinct from the surface of the perfect sphere) has the changing altitude, requiring a third number, and the concept of variable curvature of an area of the surface of the manifold associated with each distinct point.

Now while being stuck on the two-dimensional surface of the Earth, the easily verified attribute of altitude for each point on the surface gives a third dimension required to "embed" the two-dimensional surface into three-dimensions to account for the measurement/observation of altitude. Though, on a high-altitude flat geographical-plane on the surface of the Earth, the Earth is large enough that a human standing on the surface doesn't "see" the altitude nor the curvature of the Earth, but both the altitude and the curvature can be "observed" by using measurement tools.

At high relative velocities, (and velocity is measured by 4 dimensions (speed is measured by two dimensions)), the finite speed of light must be taken into consideration in making relative measurements, and the possible confusion between "seeing" and "observing" is a more entangled riddle best summarized as "Terrel Rotation":
http://www.math.ubc.ca/~cass/courses/m3 ... rell1.html
and the "curvature of axis" effect:
http://www.math.ubc.ca/~cass/courses/m3 ... ture1.html

With gravity, the 4 dimensions of space-time is "curved", which results in observable measurements of the curvature, and in which requires 5 dimensions to be embedded in for a full "intuitive" sense of the 4-dimensional manifold of the curved space-time.

One history has the confusion between "seeing" and "observing" recognized in:
http://www.guspepper.net/electro/Segund ... /Funez.pdf

and with other resulting effect examples at:
http://hep.sdu.dk/FY504/FuzzyGerry.pdf

Tadzio

The dolls can resemble a nested stack of spaces but they are not dimensions. In a true dimensional representation of a point in a space, the co-ordinates are independent of each other.

ruveyn

Ya, the internal/external size discrepancy was what I was mostly referring to.

My wife and I watched this video: [youtube]http://www.youtube.com/watch?v=IoVb6-s-wEA[/youtube]
And I asked her why, after the one-minute mark, the guy starts listing visually-obvious size discrepancies about his balls.
At one point, he compares them to the size of the moon, so I asked my wife why the lyrics were so inaccurately chosen.
She replied that his pants might be like the TARDIS.

How about the space-time continuum? and well, one variable is dependent on the other.

I don't know how it works but I presume they are trying to demonstrate that the Doctor goes into the tardis and steps into a new spatial dimension.

I've seen it played with before, it seems like a cute and edgy concept but what I don't understand is how such an object (4th dimensional) would interact with its surroundings or that the extra fourth dimension would be in a spatial sense. When I see it attempted it seems like an attempt to imply a partially inside-out factor of some type or saying that an object has some visual trickery to it - which when looked at that way would make you wonder whether its truly 4 dimensional or whether the particular visual vantage point is deceptive. The gif on the front page of the seeming cube but not a cube is a good example - where the lines line up to look like a cube but when it rotates you see that its not really a cube in there but rather just something that looks like that from a given vantage point (much like Julian Beever's sidewalk art looks 3D from the proper angle).

Only on a world-line. Not throughout the entire space. The equation for a circle makes the x co-ordinate dependent on the y or the other way around.

ruveyn

Like Predator?

That would be cool!