/sci/ - Science & Math

Confirmed it myself. For a unit square base and height [math]h[/math] the answer is the point [math]\frac{h-1}{\sqrt{2} h}[/math] along the diagonal of the top base from B.
Hints: use symmetry and reduce the problem to two contending paths; then its simple optimisation.

>>10279968
But once self-propagating systems have attained global scale, two crucial differences emerge. The first difference is in the number of individuals from among which the "fittest" are selected. Self-prop systems sufficiently big and powerful to be plausible contenders for global dominance will probably number in the dozens, or possibly in the hundreds; they certainly will not number in the millions. With so few individuals from among which to select the "fittest," it seems safe to say that the process of natural selection will be inefficient in promoting the fitness for survival of the dominant global self-prop systems. It should also be noted that among biological organisms, species that consist of a relatively small number of large individuals are more vulnerable to extinction than species that consist of a large number of small individuals. Though the analogy between biological organisms and self-propagating systems of human beings is far from perfect, still the prospect for viability of a world-system based on the dominance of a few global self-prop systems does not look encouraging.

>>10271625
The second difference is that in the absence of rapid, worldwide transportation and communication, the breakdown or the destructive action of a small-scale self-prop system has only local repercussions. Outside the limited zone where such a self-prop system has been active there will be other self-prop systems among which the process of evolution through natural selection will continue. But where rapid, worldwide transportation and communication have led to the emergence of global self-prop systems, the breakdown or the destructive action of any one such system can shake the whole world-system. Consequently, in the process of trial and error that is evolution through natural selection, it is highly probable that after only a relatively small number of "trials" resulting in "errors," the world-system will break down or will be so severely disrupted that none of the world's larger or more complex self-prop systems will be able to survive. Thus, for such self-prop systems, the trial-and-error process comes to an end; evolution through natural selection cannot continue long enough to create global self-prop systems possessing the subtle and sophisticated mechanisms that prevent destructive internal competition within complex biological organisms.

The techies may answer that even if almost all biological species are eliminated eventually, many species survive for thousands or millions of years, so maybe techies too can survive for thousands or millions of years. But when large, rapid changes occur in the environment of biological species, both the rate of appearance of new species and the rate of extinction of existing species are greatly increased. Technological progress constantly accelerates, and techies like Ray Kurzweil insist that it will soon become virtually explosive; consequently, changes come more and more rapidly, everything happens faster and faster, competition among self-prop systems becomes more and more intense, and as the process gathers speed the losers in the struggle for survival will be eliminated ever more quickly. So, on the basis of the techies' own beliefs about the exponential acceleration of technological development, it's safe to say that the life-expectancies of human-derived entities, such as man-machine hybrids and human minds uploaded into machines, will actually be quite short. The seven-hundred-year or thousand-year life-span to which some techies aspire is nothing but a pipe-dream.

>>10261327
Chaotic behavior is not limited to complex systems; in fact, some surprisingly simple systems can behave chaotically. The Encyclopaedia Britannica illustrates this with a purely mathematical example. Let A and [eqn] \ X_{0} [/eqn] be any two given numbers with 0<A<4 and [eqn] \ 0 > X_{0} < 1 [/eqn]<1, and let a sequence of numbers be generated according to the formula [eqn] x_{n+1} = Ax_{n}(1-x_{n}) [/eqn] For certain values of A, e.g., A=3.7, the sequence behaves chaotically: In order to bring about a linear increase in the number of terms of the sequence that one can predict to a reasonable approximation, one needs to achieve an exponential improvement in the accuracy of one's estimate of [eqn] \ X_{0} [/eqn] In other words, in order to predict the nth term of the sequence, one needs to know the value of x0n with an error not exceeding [eqn] 10^{-k} [/eqn] , k a constant. This is characteristic of chaotic systems generally: Any small extension of the range of prediction requires an exponential improvement in the accuracy of the data.

>>10216527
Meanwhile, fierce competition among global self-prop systems will have led to such drastic and rapid alterations in the Earth's climate, the composition of its atmosphere, the chemistry of its oceans, and so forth, that the effect on the biosphere will be devastating. In Part IV of the present chapter we will carry this line of inquiry further: We will argue that if the development of the technological world-system is allowed to proceed to its logical conclusion, then in all probability the Earth will be left a dead planet-a planet on which nothing will remain alive except, maybe, some of the simplest organisms-certain bacteria, algae, etc.-that are capable of surviving under extreme conditions.

>>10194807
Meanwhile, fierce competition among global self-prop systems will have led to such drastic and rapid alterations in the Earth's climate, the composition of its atmosphere, the chemistry of its oceans, and so forth, that the effect on the biosphere will be devastating. In Part IV of the present chapter we will carry this line of inquiry further: We will argue that if the development of the technological world-system is allowed to proceed to its logical conclusion, then in all probability the Earth will be left a dead planet-a planet on which nothing will remain alive except, maybe, some of the simplest organisms-certain bacteria, algae, etc.-that are capable of surviving under extreme conditions.