http://t.co/OYVsXcrpqm happy #towelday with @ulaulaman

### Alan Turing's declassified papers

@ulaulaman via @MathisintheAir about #AlanTuring on #arXiv
Recently Ian Taylor has uploaded on arXiv a couple of declassified papers by Alan Turing about statistics, probability and cryptography:
1. The Statistics of Repetitions
In order to be able to obtain reliable estimates of the value of given repeats we need to have information about repetition in plain language. Suppose for example that we have placed two messages together and that we find repetitions consisting of a tetragramme, two bigrammes, and fifteen single letters, and that the total overlap was 105, i.e. that the maximum possible number of repetitions which could be obtained by altering letters of the messages is 105; suppose also that the lengths of the messages are 200 and 250; in such a case what is the probability of the fit being right, no other information about the day's traffic being taken into consideration, but information about the character of the enciphered text being available in considerable quantity?
2. The Applications of Probability to Cryptography
The theory of probability may be used in cryptography with most effect when the type of cipher used is already fully understood, and it only remains to find the actual keys. It is of rather less value when one is trying to diagnose the type of cipher, but if definite rival theories about the type of cipher are suggested it may be used to decide between them.

### Inge Lehmann: the core of the Earth

http://t.co/gCOlKccELA about #IngeLehmann #EarthCore #theCore #geophysics
Inge Lehmann was a Danish seismologist and geophysicist. Using seismic data, she discovered the inner solid core of the Earth with some physical properties distinct from the outer liquid core:
No rays emerged at epicentral distances between 112° and 154°. I then placed a smaller core inside the first core and let the velocity in it be larger so that a reflection would occur when the rays through the larger core met it. After a choice of velocities in the inner core was made, a time curve was obtained, part of which appeared in the interval where there had not been any rays before. The existence of a small solid core in the innermost part of the earth was seen to result in waves emerging at distances where it had not been possible to predict their presence.

Lehmann, I. (1987). Seismology in the days of old Eos, Transactions American Geophysical Union, 68 (3) DOI: 10.1029/EO068i003p00033-02 (full paper)
Read also: Bolt, B. (1994). Inge Lehmann Physics Today, 47 (1) DOI: 10.1063/1.2808386

### The mathematics and geometry in John Hejduck

http://t.co/b67NmBNDzW about #JohnHejduck #mathematics #geometry #architecture
In the Diamond project he articulates his idea of the architect's plan as perpendicular to the observer's frontality. This makes the opposite positions between the architect and the imaginary observer equivalent, which, building on Mondrian's ideas, maximises the strength of the contrary oppositions. Hejduk's axes are no longer ordinary axes, not really comparable witb the axis of x,y and z of conventional descriptive geometry. As we turn around, the line becomes a plane. The plane becomes imagined as a wall, a wall corresponding to the bodily movement. It's the moment of passage. The outline is also a membrane, Hejduk's says, and as the relatioosbip between our present and future, the relationship between the architect and the observer, the moment of passage through the wall is a "moment of the hypotenuse", of moving from one condition to another, through an edge between two elements. The concept of the "hypotenuse" is like a cut between the equivalence, as an opening and a movement, moving across two apparently fixed conditions. Hejduk's moment of the hypotenuse, when you become physically inside, is the moment of thought appearing, memory, seeing and moving. It resembles the experience of reading a book; all of a sudden you are in it, on the inside, and it bas become a part of you. But tbere is a difference, Hejduk remarks: there is something special about the physical encounter.
from Architecture of the ineffable: on the work of John Hejduk by Einar Bjarki Malmquist