Intersection of Two Planes in Fourth Dimension
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"We may note, however, that the notion of a space of four or more dimensions, is not, as generally supposed, necessarily beyond our powers of concrete representation. True, a space of points of more than three dimensions is an abstract generalization to vizualize wich is beyond the present powers of our imagination." Encyclopaedia Britannica 1962

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The concept of Hyperspace has already appeared in the work of Georg Friedrich Riemann. But is was Curbastro Gregório Ricci who formulated the concept of Rn in a more expressive manner. Many consider him as the author and refer to “the Ricci Rn Space”. I was quite impressed when I learned about Rn in college because everything worked so well with Rn. However, Rn does not exist. Is that really so?
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Tulio Levi-Civita, one of Ricci´s pupils, went deeper into the concept of vectors and conducted thorough studies on tensors. A rather obscure matter; an issue for few. Einstein himself would have studied tensors to work out the generalized relativity theory.
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When we mention tensors, we are already speaking about Rn. .........................................................................................
Now, there are many new facts about Rn. .........................................................................................
One of the most startling discoveries is very simple. I have always used it to demonstrate that four-dimensional geometry is not merely an extension, in space, of three-dimensional geometry. .........................................................................................
To begin, we shall imagine one single point in four-dimensional space. Point O. Like any point in this space, point O shall be defined by a block of coordinates (x,y,z,w), in which x, y, z and w are real numbers. .........................................................................................
As we are in a space of four dimensions, I shall be able to build Cartesian Coordinates System X, Y, Z and W, in which each axis is orthogonal to the other three. Let us put the origin of this coordinates system in point O. There is no way to do this in our tree-dimensional space, but you can imagine a place where it is possible, or just see this with your mental eyes. .........................................................................................
Okay, that is all I need. .........................................................................................
Axes X and Y define a notable plane. Let us call it plane A. Each pair of axes defines a plane, by the simple fact that, in any space, a pair of convergent straights defines a plane. Let us consider another notable plane, the plane defined by axes Z and W. We shall name it plane B. .........................................................................................
We are going to study planes A and B because these are very special. Without any loss of generality, this theorem may be repeated with absolutely generic planes in four-dimensional spaces. The only difference is that the demonstration becomes much more difficult. .........................................................................................
All points contained in plane A, format P(x,y,0,0), i.e. their third and fourth coordinates, are zero, x and y being two real numbers whatsoever. .........................................................................................
All points of plane B are format Q(0,0,z,w), in which the first two coordinates are necessarily zero. Any number in the first coordinates, even if they are very small, means the point leaves plane ZW in the direction of the axis of the coordinate whose value increases, starting at zero. ......................................................................................... And I ask: and what about point O of the origin?
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The four coordinates of point O of the origin are zero. Its first two coordinates are zero and it belongs to plane A. However, its third and fourth coordinates are zero and it also belongs to plane B. So, surprisingly, point O belongs to both planes. As shown before, no other point of these planes satisfies both conditions, thus point O is SINGLE. .........................................................................................
As the reverse straights are a geometrical fact that is unknown in the plane (two dimensions), intersection of planes in one single point is a fact that has been absolutely unknown to us. All pairs of planes we know are either parallel or intersect, following a straight line. Observe the intersection lines of the planes of the room where you are. Wall and ceiling, wall and floor, for example. Ceiling and floor are usually parallel. ......................................................................................... To finish this off, I ask you: how is the intersection of the plane defined by axes X and Y and the plane defined by axes X and W?
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If you can answer this question, accept my congratulations. .........................................................................................
You are already beginning to learn Hypergeometry although I have not even drawn any figures. .........................................................................................
Writing this paper has thrilled me overwhelmingly. .........................................................................................
In vain have I tried to publish this paper in the best scientific magazines of Brazil and abroad. In vain. .........................................................................................
And now, there you are, surfing my site, like a proud swan that used to be an ugly little duck in the past. .........................................................................................
In spite of its simplicity, this work shows us the wealth of Hyperspace. In a plane, two straights are either concurrent or parallel. In space, the increase of dimensions opens new possibilities. Now, that is what this paper is meant for: show how important it is to study Hypergeometry as a tool to explore new possibilities. .........................................................................................
Okay, in four dimensions we have spherones. Spherones are geometrical entities which, considering three dimensions only, may be cones or spheres (at the same time). ......................................................................................... Has any physicist ever seen more delicious tidbits ?
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Have fun, friends. Through this site, Hypergeometry and Hyperphysics have been set free!
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