When we stack cannon balls such that each one touches all the others immediately adjacent to it we can place them in several different configurations. One is to place four in a square pattern and a fifth on top in the middle of the four. We can repeat the pattern to the sides and in additional layers. This produces a square pyramid. Another is to place three cannon balls in a triangle and a fourth on top in the middle of the three. We can also repeat this pattern to the side and in additional layers. This makes a triangular pyramid.
I find the triangular pyramid more interesting than the square pyramid. If we imagine lines connecting the centers of the first four cannon balls we recognize a tetrahedron, a shape with four vertices and four faces which are equilateral triangles . We can produce a larger tetrahedron by adding more cannon balls so that the number of the layers is equal to the number of cannon balls along the edges.
The interesting part, to me, is that if we had placed a canon ball just below ground level and then placed three in a triangle around that one and a fifth one on top of the three we could imagine a shape with five vertices and all six faces would be equilateral triangles. This is called a triangular bi-pyramid.
We should recognize that a triangular bi-pyramid tessellates a volume. If we consider that the vertices imagined in the three cannon balls at ground level form a plane and all six of the other faces form planes we have the basis for a seven-plane geometry of a volume in the same fashion that tessellation of a volume by the cube provides the basis for a three dimensional geometry.
Show your friends that the universe is actually not 3-D, but really is 7-D.
As they say on TV: don’t try this at home. At least not with cannon balls. Cannon balls are heavy, expensive and can be dangerous.
Better to use ping-pong balls. They are light, inexpensive and available in colors. Just be sure to get good ones, very round and rigid. It’s easier if you use a box to contain the ping pong balls.
Crafty folks may want to form the geometric solid by cutting out the net; add tabs for gluing. The solids may not be as intuitive as the spheres for demonstrating the 7-plane geometry.
I believe the correct answer for how many physical dimensions the universe has is one. That dimension is volume. I think that physical volume cannot be shown to be a reduced form of something else (but I will keep an open mind to that) nor can physical volume be reduced to what we call area or length.
It is very commonplace for “dimensions” to be seen as this mysterious thing where strange aliens may hide, as in “alternate dimensions”. There are 3, for certain, unquestioned dimensions. They are height, width and depth. Volume is a measurement accounting for all 3 simultaneously.
The existence of a volume must precede an attempt to describe it.
The three dimensions you identify are descriptions of the intersections of the planes in three plane geometry. That geometry arises because the cube, a basic geometric shape, can be used to fill (tessellate) uniformly a volume (a space) with replicas of that cube. There are several accepted geometries for identifying places in a space. Polar geometry and cylindrical geometry are more useful than cubic (3-D) geometry in some cases. Three planes (3-D) are accepted as the minimum number required to describe places in a volume, but the polar, cylindrical and 7-D geometries are equally valid.
I have identified another basic geometric shape, the triangular bi-pyramid, which also tessellates a volume. It gives rise to a seven plane geometry (7-D). 7-D is much more accurate than the others when used to describe a space filled with spheres or a space filled with spheres so small that they approximate a point.
7-D provides integer values for the position of each sphere, or approximated point. Integer values are particularly useful when describing the least energy, or rest state, of a volume filled with uniform objects such as the smallest possible object or elementary particles.
I sincerely hope you are having us on. But it is not bad for a third grade introduction to the world around us.
I’d like a square foot of coffee please, with an inch of cream.
I assume this is for me. I used an elementary link because it’s an elementary concept. Terms being critical. You gave a more complex definition about 7-D, which seems to be out of whack with your previous 1-D, and none of which sheds any light on whatever it is you are talking about.
Lausten: “I assume this is for me. I used an elementary link because it’s an elementary concept. Terms being critical. You gave a more complex definition about 7-D, which seems to be out of whack with your previous 1-D, and none of which sheds any light on whatever it is you are talking about.”
Yes, I suppose my original post was a bit cheesy, maybe more than a bit. I encounter many people who apparently are convinced that the universe has three dimensions that are fundamental physical attributes. Their up-down, forward-backward, left-right experience on the surface of the Earth seems to limit their ability to grasp the true nature of the universe.
I tried to show, with the visual aid of stacked cannon balls (ping-pong balls), that the common 3-D geometry is just one of many possible geometries we can use to describe places, or positions, in the volume we call space.
Since I missed the mark with you I have to assume I missed it with everyone; my lighthearted approach failed. However, I think that your reference to “it” (a 3-D world) as an “elementary concept” reinforces my position.
It was a South Park joke. Geometry is not physics. Physics is physics. Yes, there is math in physics which tells us things, or at least suggests them. Math brought us big bang theory. Math brought us 11 dimensions. But not math alone, ever. Math informed by observations of the universe brought us big bang theory. Math informed by observations of the universe brought us 11 dimensions. Math alone is not enough to tell us anything about the universe. It must be math informed by the universe itself.
I have never seen South Park. I suppose that was obvious.
I’ve been waiting on someone to tell me that seven-plane geometry is not 7-D and that it would be better to call it 6-D since the intersections of the seven planes produce only six of the lines that commonly would be called dimensions (in the geometric sense). I suspect there is a rule for that somewhere, but I haven’t found it. I’m still not sure which one conveys the meaning best.
I would not attempt to do physics in a pseudoscience arena. The observation of a geometry inherent in the ways we stack cannon balls and the fact that a triangular bi-pyramid tessellates a volume leads me to believe it has application in physics. I have noted in particular that it can describe in integer terms the rest state of least energy stacked spheres. This could be a basis for a description of the state of the universe at least energy or zero energy state. I think this qualifies as pseudoscience.
Okay, I think I understand your point a little better now and believe I have an intelligent question to ask. If I understand you correctly, this math doesn’t actually say, “There ARE 7 dimensions”. It actually says, “There are AT LEAST 7 dimensions”. After all, being able to draw on a page does not say that there ARE 3 dimensions, just that there are AT LEAST 3 dimensions, those being the 2 we are drawing on and the third which separates us from the page.
I guess it all depends on the definition of “dimension”. In a general sense, a dimension could be any characteristic, attribute or quality of an object, event or process. I was trying to stick to the definition in a geometric sense where any position in the volume is described as an offset from an origin.
In the common 3-D geometry where the origin is P(0) and the position to be described is P(1):
P(1) = P(0) + (aX, bY, cZ) where a,b and c are units along the X, Y and Z axes respectively. All the other math in this geometry is developed from this equation. These three axes provide all of the 3-D geometric dimensions; there are no others.
In the same way in 6-D geometry (I have decided to use 6-D rather than 7-D to be more conventional. I should have had the sense to do that to begin with.):
P(1) = P(0) + (aL, bM, cR, dS, eT, fW) where a, b, c, d, e, and f are units along the L, M, R, S, T and W axes respectively. All the other math in this geometry is developed from this equation. These six axes provide all of the 6-D geometric dimensions; there are no others.
In both the common 3-D and this 6-D the letters are just labels chosen to be able to keep up with axis is which. There is no other meaning.