The Arecibo message pictured above was beamed into space via frequency modulated radio waves at a ceremony to mark the remodeling of the Arecibo radio telescope on 16 November 1974. It was aimed at the globular star cluster M13 some 25,000 light years away. The message consisted of 1679 binary digits.
The cardinality of 1679 was chosen because it is a semiprime (the product of two prime numbers), to be arranged rectangularly as 73 rows by 23 columns. The alternative arrangement, 23 rows by 73 columns, produces jumbled nonsense.
Knowledge of how to read and write Egyptian Hieroglyphics had been forgotten since shortly before the time of the fall of the Roman Empire. It was the finding of the Rosetta Stone (a piece of what had been a "stele" engraved with a decree) that allowed Egyptian Hieroglyphics to be deciphered. The stone was discovered in 1799 by a soldier in Napoleon's army. When Napoleon went on campaign in Egypt he brought scientists with him, and they recognized the importance of this find. The stone had an inscription written in three languages: Egyptian Hieroglyphics, Demotic Text (also called "ancient cursive Coptic"), and Ancient Greek. The Ancient Greek language was known (in fact until a couple of decades ago any truly educated person learned Ancient Greek and also Latin). The fact that these three languages were expressing the same decree and that one of them was well-understood became the key to unlocking knowledge and understanding that had been lost for more than a thousand years. Here is the stone, which is now housed in the British Museum in London:
And here I am looking at the stone in 2003 when I had opportunity to travel to England. Of all the things in London that there were to see, this was definitely on my "Top Ten" list!
Here are the first few stages of the construction of the HILBERT CURVE. Notice that it begins as a one-dimensional segment. If we think of dimension as how much information you need to locate a point, we see we only need one piece here - how far along the curve we are. If iteration process is carried out infinitely many times the curve contains every point in the square; in other words it fills the square. But we know that a square is two-dimension, so what is this shape: one-dimensional, two-dimensional, both, neither?
Below is the CANTOR SET. It is created by beginning with a segment (one-dimensional), cutting it into three pieces and removing the middle third. This process is repeated at the next stage for all newly created segments. The final result is an object with no length, made up of intinitely many points. We know points are zero-dimensional. What is the dimensionality of the Cantor Set - one-dimensional (like the shape it began as), zero-dimensional (as a point is), both, neither?
Next we have the KOCH CURVE, in which we begin with a line segment, cut it into thirds and replace the middle third by two segments of the same length. At each new stage we do this for each new segment. In this way we get a curve in which the endpoints never get further apart, and which we could surround completely with a small oval, and yet that has infinite length. Also, any tiny piece of the final shape can be blown up to give us an EXACT copy of the original (see animation zoom).