A 3D-sphere mapping?

Higher-dimensional geometry (previously "Polyshapes").

A 3D-sphere mapping?

Postby begvend » Fri Apr 17, 2009 7:32 pm

Hello all,
On the 2Dsphere there are:
- 1 parallel.
- 2 meridians.
In this case:
- A 2Dsphere is mapped with 8 spherical triangles (polygon).
- A polyhedral simplification is octahedron.

On the 3Dsphere there are: (I suppose)
- 1 parallel.
- 2 meridians.
- 3 Hyper-meridians.
In this case, the 3Dsphere is mapped with polychora.
- What are types and quantity of these polychora?
- What is polychoral simplification?

Thanks for your comments.
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Re: A 3D-sphere mapping?

Postby wendy » Sat Apr 18, 2009 5:06 am

Glome ("3d sphere")-mapping, including meridians etc, are more relevant to the notion of planets and suns in space, as well as the idea of rotation of the planet.

Assuming that the planet rotates in the Clifford-manner (that is, every point around the centre: the most stable manner), and that the sun tracks around this in an oblique manner (the general case), then, the following applies.

1. There are time zones, according to as the point sees the sun. This is like longitude.

2. There are season-zones. For this, think of the Earth seasons as a circle as a spinner (with N, S), on a full year. As time goes by, S points to the seasons in order, and the N follows six months behind. In four dimensions, the spinner is replaced by a full disk. This is like our N/S distinction, except that it is now a full angular thing. (somewhere there must be spring).

3. The third coordinate corresponds to near or far from the sun, roughly like our lattitude. Near the sun gives the tropics, or hot, while away from the sun gives the artics, or cold.

Since the nature of maps is to show different things that can be related to the above, we can start doing some 'division'.

Dividing the sphere by longitude, gives a half-sphere, like the gimble that holds a globe. On the glome, the division forms a half-sphere, which, by doubling the distance from the point opposite the cut, gives the 'zenith sphere'. This is interesting, since it is the same for everyone on the planet. One simply sits it so that the zenith is at the top, and the point opposite is the stars that hug the horizon.

On the sphere, one sometimes sees the Zodiac, or sun-track, traced out such that a full circle is a year. It looks like a sine wave. The idea is to show what lattitude the sun is in at any month, In practice, one maps this onto the gimble, and the whole line of lattitude is variously under the sun at that time of year.

In four dimensions, the Zodiac appears on the Zenith-sphere as a circle. This circle is a kind of year-lattitude, the length of the circle divided into year-longitude. The sun travels around the year-longitude line in one year, the particular angle it falls in is 'midsommer' there.

Taking this line on the Zenith-sphere as a zennith-lattitude, we see that the zenith-lomgitude corresponds to all parts of the world that is having that particular season (at a day-to-day level). The S pole corresponds to a sun-line, where the sun rises to exactly the same height every day, while the N pole the sun does the same, but never gets high in the sky.

Were you to mark this zodiac-circle with "standard months" (i to xii), such that everywhere it is say vii or whatever. You now place such zodiac sphere on the table such that the top is the zenith-star that matches your location. The top of the circle is the month that mid-summer falls, the bottom where mid-winter falls. You can see that mid-summer could for example, fall in vii or ix or i or whatever. This also tells you how far the sun climbs in the sky on that particular day.

On the ground, the E/W (sun-direction) presents as a line that runs from the point of rising to setting of the zenith star. The horizon is in the shape of a sphere, where the E and W poles are the rise and set points. The zodiac appears on the zenith-sphere as a circle, is projected onto the E hemisphere by lying the sphere flat on the ground, that the zenith points E, and the nadir points west. You stand at the nadir of the Zenith sphere, and the line through any given point on this zenith-sphere strikes the horizon where that star rises.

The NS axis is perpendicular to EW axis, but then we see that perpendicular to EW we have a full circle.
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