Triangles on Earth Exceed 180 Degrees

The idea that space and time can be curved or warped is fairly recent. For more than 2,000 years the axioms of Euclidean geometry were considered to be selfevident. As those of you who were forced to learn geometry at school may remember, one of the consequences of these axioms is that the angles ot a triangle add up to 180 degrees. However, in the last century people began to realise that other forms of geometry were possible in which the angles of a triangle need not add up to i8o degrees...

 A Library is Not a Cathedral

"Crazy?" He rounded on her, all angles and motion, and shook a finger under her nose. "I know your type and your pathetic delusions. Oh, yes, I do. You think of a library as being like the mind of a great and noble Scholar—catholic, universally educated, and precisely organized. Every opinion balanced against its opposite, every fact quickly retrievable. The only biases those that exist in the knowledge itself. If a gap exists in the collective omniscience, a horde of servants will scurry t...

 A Library is Not a Cathedral

"Crazy?" He rounded on her, all angles and motion, and shook a finger under her nose. "I know your type and your pathetic delusions. Oh, yes, I do. You think of a library as being like the mind of a great and noble Scholar—catholic, universally educated, and precisely organized. Every opinion balanced against its opposite, every fact quickly retrievable. The only biases those that exist in the knowledge itself. If a gap exists in the collective omniscience, a horde of servants will scurry t...

 Knowing the World Requires Mathematics

For the things of this world cannot be made known without a knowledge of mathematics. For this is an assured fact in regard to celestial things, siDce two important sciences of mathematics treat of them, namely theoretical astrology and practical astrology. The first. . . gives us definite information as to the number of the heavens and of the stars, whose size can be comprehended by means of instruments, and the shapes of all and their magnitudes and distances from the earth, and thicknesses...

 Dissecting Crystals

A casual glance at crystals may lead to the idea that they were pure sports of nature, but this is simply an elegant way of declaring one's ignorance. With a thoughtful examination of them, we discover laws of arrangement. With the help of these, calculation portrays and links up the observed results. How variable and at the same time how precise and regular are these laws! How simple they are ordinarily, without losing anything of their significance! The theory which has served to develop th...

 Cyberspace as a State of Mind

Occasionally, the metaphoric location of the mind of a person in hack mode. Some hackers report experiencing strong synesthetic imagery when in hack mode, interestingly, independent reports from multiple sources suggest that there are common features to the experience. In particular, the dominant colors of this subjective cyberspace are often gray and silver, and the imagery often involves constellations of marching dots, elaborate shifting patterns of lines and angles, or moir