20 March 2012

Equinox, or how to define things properly

Google reminded me of Equinox today.
Equinox is defined as a point on the trajectory (and hence a point in time) of the Earth around the Sun at which the line L joining the Sun and the Earth and the axis A around which the Earth revolves are perpendicular to one another. Since A remains, approximately, fixed in space, it turns out that there are exactly two equinoxes (a simple consequence of the intermediate value theorem for continuous functions).

Compare this simple definition with the one given on Wikipedia:

An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator. The term equinox can also be used in a broader sense, meaning the date when such a passage happens. The name "equinox" is derived from the Latin aequus (equal) and nox (night), because around the equinox, the night and day have approximately equal length.
At an equinox, the Sun is at one of two opposite points on the celestial sphere where the celestial equator (i.e. declination 0) and ecliptic intersect. These points of intersection are called equinoctial points: classically, thevernal point and the autumnal point. By extension, the term equinox may denote an equinoctial point.
An equinox happens each year at two specific moments in time (rather than two whole days), when there is a location (the subsolar point) on the Earth's equator, where the center of the Sun can be observed to be vertically overhead, occurring around March 20 and September 22 each year.

Such a simple concept, but such a convoluted definition. No wonder that many people have little understanding of trivial facts, such as the equinox.

Of course, we should not forget that the definition above will not satisfy some creationists, for whom the Earth does not move, because--they claim--the Vatican believes the same thing, and because their religious texts say so (blessed are the poor in spirit, for theirs is the kingdom of heaven; Matthew 5:3).

P.S. The Earth's rotational axis, denote by A above, is not really fixed but moves very very slowly. So slowly that it takes 26 thousand years to complete a cycle. Today, A points towards the star Polaris (the commonly known Northern Star), but 10 thousand years ago it did not. This phenomenon is known as precession of the equinoxes because, as a result of it, the equinoxes change very very slowly too. It was described by the ancient astronomer Ptolemy, about 2000 years ago, who attributed it to Hipparchus who was born 200 years before Prolemy.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant