Date: Sep 1, 2005 11:21 PM
Author: Kirby Urner
Subject: More Math/CS Visualizations

Draft curriculum writing, this and final to be linked from
Another Alien Curriculum, first published to the Math Forum and positively reviewed by Hake.

http://www.4dsolutions.net/ocn/alien.html

If you find errors, feel free to append and/or contact the
author, Kirby Urner (urner@alumni.princeton.edu). That's
in the next few weeks. Presume this material has been
superceded if archival (e.g. by the final draft).

Kirby

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Given the advent of GIS/GPS ala Google, handheld devices etc., plus jet travel and events impacting other events globally, the curriculum segment on coordinate systems should place a lot of emphasis on latitude/longitude. This is in keeping with my approach of starting with spatial geometry and resolving to the plane at a later time, when said plane has already been contextualized by explorations involving polyhedra and their relative volumes in a concentric hierarchy, which hierarchy is in turn embedded in a lattice coincident with dense packed spheres.

Logistically speaking, we want our students to see the stars, which is a non-trivial request in densely populated areas full of light pollution. A planetarium or computerized star field (projected) has conceptual value and a lot of capabilities, but there's no replacement for an actual experience of the night sky, untainted by much local lighting.

Getting out on the open ocean is also a plus, or at least onto a flat plain relatively unmarred by topography. The philosophy here is that we're grooming global citizens who have a good sense of what the planet is like, and who treasure it, want to pass it on in good shape. Stewards of the Earth should be empowered to actually experience the Earth. An education system concerns itself with the logistics of making this happen. Some necessary jobs requiring not a lot of skills should be earmarked for traveling students in our global university system. Students could book time with an income, plus have time left over to take in their new environments.

Such systems will be a worthy investment, both in our students and in the long term health of our ecosystems, all of which are impacted by human decision making and activities.

Out on the ocean, or on a vast plain, the circle of the horizon becomes more evident, as a kind of flat disk. From earlier studies, students will see this azimuth-bounded disk as tangent to the sphere of the Earth, a patch of real estate that bounded by the curve of the planet's surface. Clouds that appear lower in the sky at the rim are actually at the same altitude -- it's just that the sky itself is bending around in all circumferential directions.

The point directly overhead is called the zenith. It's perpendicular to the disk bounded by the azimuth. We might say it describes a normal vector (normal being a synonym for perpendicular). In the direction opposite the sky, this vector goes to the center of the Earth, there to form angles with other vectors on other planes.

A geoscope is a transparent ball with geodesics etched onto it, helping to orient the observer at its center. One of these geodesics corresponds to the equator of the Earth, i.e. is parallel to the plane of the equator. Another geodesic is the ecliptic, which marks the path of the sun. The local horizon maps to the geosphere as yet another geodesic. If one happens to be at either pole, one's actual azimuth runs parallel to the equator. Otherwise, it tilts relative to the equator, and if imagined as a geodesic, intersects the equator in two places, as does the ecliptic.

As the horizon disk rotates around a center point at the
center of the geosphere (the viewpoint of the observer),
the sun appears to arc across the sky, until cut off from
view by the western lip of the horizon, rolling away
from the sun as the planet turns. The ecliptic is not a
constant relative to the horizon, i.e. its angle to that
of the local disk is not fixed throughout the year, nor
are the rising and setting points a constant.

In the northern hemisphere, the sun arcs lower in the winter sky, and in the far north only appears briefly, if at all. Around the winter solstice, the southern hemisphere enjoys the highest ecliptic of the year, and the warmest days -- it's summer in other words. The seasons rotate 180 degrees out of phase. As the north is passing through fall enroute to winter, the south is passing through spring enroute to summer. Migratory birds and other species, many humans, take advantage of this phase shift to stay in warmer climes throughout the year [suggested viewing: Winged Migration, film, http://imdb.com/title/tt0301727/].

When not facing the sun, we see the other stars. Given the planet is rotating around the sun, the vista outward from the sun is always changing, always repeating. The set of constellations in the direction of the sun (forming its background) are what the sun is "in". The constellations 180 degrees away from the sun reach their highest point relative to one's zenith as one's viewpoint revolves to a point opposite the sun. For example, during the March equinox, the sun is "in" Pisces, and facing outward towards Virgo.

The new constellations appear towards the east and arc westward as the year progresses. The constellations one sees has everything to do with the local horizon disk. Some will be too far north or south. The trajectories of the constellations also depends on the orientation of the local disk. Some will arc directly overhead, while others may follow a circular path (when not obscured by sunlight) in the extreme south or extreme north.

Children living in a high tech ecovillage may have a
community geoscope available as part of their schooling.
Planetaria may offer similar services to city kids. The
planes of the equator, ecliptic, local horizon need to
be shown against the backdrop of the night sky, as well
as during the day. The fourth plane of interest: that
of the galaxy itself. The Milky Way galaxy forms a disk
with its own geodesic.

With these four planes established, and their relationships understood, it becomes easier to locate and discuss various objects, both in the local solar system, and in the distant reaches of space. The direction of the sun's motion, relative to local stars and the galaxy, may be discussed. The moon, its trajectory, its phases, as companion to the Earth, will make more sense given the conceptual apparatus defined by these four planes. The paths of the planets, space probes, and orbiting satellites (including geostationary) may be investigated against this backdrop. Finally, the constellations themselves may be demarcated (and used to discuss positions), as a hodge podge of regions, as historically circumstantial and arbitrary as the political boundaries on the earth's surface. The geoscope may be used to illumine these boundaries. Each constellation/region occupies a number of "square degrees" -- unit triangles may also be used.

Mathematically, we have a lot of threads going. In terms of software engineering, we have a model-view-controller (MVC) infrastructure to discuss. At the more elementary level, we have intersecting disks, vectors, geodesics, spherical coordinates, latitude and longitude. Time calculations enter in to this picture, as do calculations involving the speed of light (1 Parsec = 3.08568025 * 10**16 meters). Astronomy will make a lot more sense, with all these mathematical seeds planeted, and nurtured thoughout the student's career.

Primary references:
Guy Ottewell. The Astronomical Companion. 1979. ISBN 0-93456-01-0

Re Geoscope:
R. Buckminster Fuller. Critical Path. 1981. ISBN 0-312-17488-8