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A parenthesis: Space-time orientation.
** ...**In the 2D world, we had equated geometric objects to letters. In the 3D world, they had been equated to "right hand" and "left hand."
Four-dimensional structures had been equated to animated holograms.
**...**What could a five-dimensional structure, or a ten-dimensional one, possibly be? Sometimes I envy God, don't you?
**...**He must be laughing at our pitiful four-dimensional structures.
**...**But a theoretical physicist, and even a mathematician, is nothing more than an oriented four-dimensional structure. If they weren't oriented this way, they couldn't distinguish past from future, nor right from left.
**...**The universe as a whole is a four-dimensional structure. Let us imagine it as a closed object with locally spherical topology. Call t the time. At a given moment, we can make a slice, which is a 3D hypersurface. If this hypersurface is a hypersphere S³, time has meaning. The time vector crosses this hypersurface, and we avoid any paradoxical situation.
**...**Let us reduce the number of dimensions. Imagine a closed two-dimensional world, a kind of space-time (x, y, t).
**...**We can slice it at t = constant, obtaining a geometric object whose dimension is 3 − 1 = 2: a 2D surface. At every point, the oriented normal vector represents the arrow of time.
If this space-time can be oriented in time (we assume it is closed), then space is a sphere S²:
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**...**But suppose the surface representing space is single-sided. Take, for example, a Boy’s surface (a closed single-sided surface; see the "Mathematics" section of the site).
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You can construct one by gluing together Möbius strips. I show you one:
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You know it's impossible to define an oriented normal vector:
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**...**The double cover of a Boy’s surface is a sphere S². If we equate our three-dimensional space-time to a set of S² spheres arranged like Russian dolls, each corresponding to a specific value of cosmic time t, we can barely imagine a certain type of space-time where antipodal points might be identified. This was the topological structure suggested in the paper:
Jean-Pierre Petit: "The Missing Mass Problem." Il Nuovo Cimento B, vol. 109, July 1994, pp. 697–710.
**...**Then we know that antipodal points located on a "meridian" of a sphere can be arranged as the double cover of a Möbius strip:
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We thus see how spatial antipodal regions are conjugated with opposite arrows of time.
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**...**Incidentally, we see how this would conjugate enantiomorphic objects.
**...**Space is a four-dimensional hypersurface. If we can define a cosmic time t, we can make slices at t = constant, and these slices are three-dimensional spaces. If space is closed, we could equate it to a sphere S³, which can be modeled as the double cover of a projective space P³ (the 3D equivalent of a Boy’s surface). This operation would bring regions with opposite arrows of time into interaction.