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To date, there are only five recognized Platonic solids: The tetrahedron, the cube, the octahedron, the icosahedron, and the dodecahedron. Why not a five-sided Platonic solid? But, if found or discovered, then what would be the benefit?
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There is another issue of interest with your challenging question, Prof. Scott Mavers,
In crystal physics, we know about 1-fold, 2-fold, 3-fold 4-fold, and 6-fold symetries. But what about 5-fold symmetry?
n-fold refers to the way an axis is rotated to get the same structure (n = 1,2,3,4 or 6) please refer to * for a very pedagogical introduction, but n = 5? Well, n = 5 are the so-called quasicrystals widely mentioned, in such a case, Prof. Kepler tried the so-called Kepler’s Pentagon Tiling **
Best Regards.
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The title needs no elaboration, I feel.
A more specific secondary question is this. If we do choose to take seriously the all-embracing creation theory postulated in Timaeus, what does that theory imply for the future of Artificial Intelligence systems?
Note: Timaeus is world class philosophy, and multiple sources and descriptions for it can easily be found on the internet. I use the Waterfield translation in the series Oxford World's Classics, published by Oxford University Press.
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Dear Professor James Doran ,
I thank you very much for your proposing this discussion.
It is not easy to give an answer: I think that the Timaeus is and will remain of interest for interpreting Plato's strategy, since in Timaeus many elements of Plato's vision of reality are exposed. Plato's positions can be interpreted as a beginning of an intelligent design assumption.
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How can the number of Ru atoms on the surface of a Ru nanoparticle (say 2 nm, hcp) be estimated? I know about the magic number approach but I am not sure which platonic system does hcp Ru belong to.
Can someone help?
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At the same time the crystal structure of Ruthenium is hexagonal close-packed (hcp) but in the above-mentioned paper presented the equations for number of atoms for cubic lattices.
Maybe it will be better to calculate the surface of nanoparticle and divide it on the surface of atom (π*r{Covalent radius}2).
With kind regards,
Liliya
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Introduction to Sequences
Obviously, there are an infinite number of possible sequences, and thousands have been studied. Some sequences contain deep insights into number theory. A sequence could be either infinite or finite. For simplicity, I will only consider sequences where the entries are real numbers (or infinity), although, you could have a sequence where the entries are any kind of mathematical object whatsoever.
A sequence has entries
a_1, a_2, a_3, a_4,...
One way to determine the entries, is to define a function f(n), and then evaluate it for positive integer values of n.
a_n = f(n)
f(1), f(2), f(3), f(4), ...
I. Sequences From a Function
1. Constant function f(n) = c
c, c, c, c, ...
2. Integers
f(n) = n
1, 2, 3, 4,...
3. Even Numbers
f(n) = 2n
2, 4, 6, 8,...
4. Odd Numbers
f(n) = 2n - 1
1, 3, 5, 7,...
5. Counting by c's
f(n) = cn
c, 2c, 3c, 4c,...
So, for example, "counting by 10's" would be 10, 20, 30, 40,...
6. Perfect Squares
f(n) = n^2
1, 4, 9, 16, 25,...
7. Powers of 10
f(n) = 10^n
10, 100, 1000, 10000,...
8. Alternating Sequence
f(n) = (-1)^n
-1, 1, -1, 1,...
II. Sequences where Entries are Derived from Previous Entries
9. Doubling Sequence
a_(n+1) = 2a_n
1, 2, 4, 8, ...
10. Halving Sequence
a_(n+1) = a_n/2
1, 1/2, 1/4, 1/8,...
11. Fibonacci Sequence = each term is the sum of the previous two terms
a_(n+2) = a_n + a_(n+1)
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,...
If a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates the Golden Ratio, φ = a/b = (a+b)/a = 1.618033988.... The Fibonacci Sequence shows up repeatedly in botany. The number of petals of flowers are almost always Fibonacci numbers.
III. Sequences Related to Primes
12. Primes = positive integers greater than 1 that has no positive integer divisors other than 1 and itself
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,...
13. Composites = positive integers that are not prime
4, 6, 8, 9, 10, 12, 14,...
14. Prime Counting Function
pi(x) = number of primes less than x
0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6,...
15. Prime Sums
sigma(x) = sum of all primes less than x
2, 5, 10, 17, 28, 41, 58, 77,...
16. Twin Primes = primes where the nearest prime is 2 away
3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43,...
or making it explicit what the pairs are
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43),...
17. Cousin Primes = primes where the nearest prime is 4 away
18. Sexy Primes = primes where the nearest prime is 6 away
19. Prime Gaps = differences between consecutive primes
1, 2, 2, 4, 2, 4, 2, 4, 6, 2,...
20. Mersenne Primes = a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n
3, 7, 31, 127,...
21. Fermat Numbers = numbers of the form 2^2^n + 1
3, 5, 17, 257, 65537, 4294967297, 18446744073709551617,...
22. Fermat Primes = Fermat number that is prime
3, 5, 17, 257, 65537
It is believed that these are the only five terms in the sequence.
23. Pseudoprimes = A pseudoprime is a composite number that passes a test or sequence of tests that fail for most composite numbers. There are actually different types of pseudoprimes based on what tests it passes or what qualities it shares with prime numbers. Some examples of pseudoprimes are Fermat pseudoprimes, Poulet numbers, Euler-Jacobi pseudoprimes, strong pseudoprimes, and Carmichael numbers.
24. Perfect Numbers = positive integers that are equal to the sum of their proper divisors
6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128,...
25. Amicable Numbers = one of a pair of positive integers where each is equal to the sum of their proper divisors of the other
220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595,...
or making it explicit what the pairs are
(220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595),...
IV. Other Sequences
26. Hiccup Sequence = integers with an extra "1"
1, 1, 2, 3, 4,...
27. Asymptotic to Pi
3, 3.1, 3.14, 3.145, 3.14159,...
28. Look Up and Say
1, 11, 21, 1211, 111221, 312211, 13112221,...
You read the numbers out loud. When you see "1", you say there is "one one". When you see "11", you say there are "two ones", etc.
29. Palindrome Numbers
1, 2, 3, 4,... other examples include 11, 22, 33, 101, 121, 424, 1001, 246642, 332233, 100040001, etc.
30. Number of Regular Examples of Polytopes in Each Dimension
1, 1, infinity, 5, 6, 3, 3, 3,...
The only polytope in zero dimensions is the point. The only polytope in one dimension is the line segment. Both could be considered regular. In two dimensions, there are an infinite number of regular polygons. In three dimensions, there are five regular polyhedra, which are the five platonic solids, the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. In the four dimensions, you have 4D analogs of each of these, plus an extra one called the 24-cell. In five or more dimensions, you only have analogs of the tetrahedron, cube, and octahedron.
31. Dimensions of Normed Division Algebras
1, 2, 4, 8
There are only four normed division algebras. The real numbers are one dimensional. The complex numbers are two dimensional. The quaternions are four dimensional. The octonions are eight dimensional.
This example shows that you can not guess what the next term is in a sequence, or even if there is a next term, from simply looking at the first few terms. If you looked at the above sequence, you might incorrectly guess that the next term was 16, when, in reality, the correct answer is that there is no next term because it is a finite sequence with only four terms.
In Elementary school, a student might be asked to write down the next few terms in the sequence
5, 10, 15, 20,...
The teacher expects you to write
5, 10, 15, 20, 25, 30, 35, 40,...
If the student wrote
5, 10, 15, 20, 1, 2, 3, 4,...
the teacher would mark it wrong. However, that is a completely valid answer. That is just as much an acceptable sequence as the one the teacher was looking for. The teacher might say, "Assume the pattern holds". Well, from just looking at the first four terms, we do not know what the pattern is. Maybe the pattern is this.
5, 10, 15, 20, 1, 2, 3, 4, 5, 10, 15, 20, 1, 2, 3, 4, 5, 10, 15, 20, 1, 2, 3, 4,...
The truth is, in order to calculate the terms in a sequence, you must be given the definition of the sequence. The first few terms alone give zero indication of what the next term will be.
Often on the Internet, you see so-called puzzles where you are supposed to guess what comes next in the sequence. Of course, from the point of view of mathematics, any answer is equally valid, whether or not it is what the writer had in mind. It not a test of mathematics. It is a test in psychology, trying to guess what the writer probably meant.
There are sequences where we do not know whether it has zero terms or an infinite number of terms. Consider the following sequence.
32. Exceptions to the Riemann hypothesis
If the Riemann hypothesis is true, then there are zero terms in this sequence. If it is not true, then there are an infinite number of terms in this sequence. Right now, we do not know which it is.
33. Coefficients of Modular Function j
1, 744, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600,...
34. Degrees of irreducible representations of Monster group M
1, 196883, 21296876, 842609326, 18538750076, 19360062527, 293553734298, 3879214937598, 36173193327999, 125510727015275, 190292345709543,...
The above two sequences are related to Monstrous Moonshine. In November 1978, J. McKay noticed that the [q bar] -coefficient 196884 is exactly one more than the smallest dimension of nontrivial representations of the Monster Group (Conway and Norton 1979). In fact, it turns out that the Fourier coefficients of j(tau) can be expressed as linear combinations of these dimensions with small coefficients.
This was a very surprising connection between two seemingly unrelated branches of mathematics. It turned out to have deep connections to string theory. The name "Moonshine" was a reference to the phrase "talking moonshine" meaning saying something untrue, although in this case, it did turn out to be true.
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What issue re you pointing at? Convergence?
What a collection!
In inteligence tests you are supposed to see a pattern, not anything goes.
If the test is not good, perhaps we can interpret the pattern of a sequence in more than one way.
Rings instead of normed division algebra can be anything dimensional. Complex nos. just one example.
Try something like x+ty+ttz with ttt=1 or -1
or x+ty with tt=0 or 1
No norm here, just a determinant of a representation. Cannot divide by any nonzero element, but by most.
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Organisational change authors often refer to the famous Heraclitus fragment, 'Everything flows. Nothing remains the same.' I am interested in identifying references to Heraclitus and other presocratic philosophers in published work on organisational change. These may relate to the above fragment, either drawing on the Platonic interpretation that this means that the world is in a constant state of change, or on alternative interpretations. There may also be references to other fragments of Heraclitus, such as those related to his theory of opposites.
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If you see the world as something constantly changing, you will be unable to to grasp the nature of things. Greek philosophers believe that time is a constant line of events. However this view is object-centered, meaning, that here we just take a look at the object and from this we try to say something about the history, present state and future. This is not a good idea and it tells us nothing about correlations and choices which have been made.
Events are rather like fields of influences. A plant only grows at daytime and when night comes the plant rests. When we only look at the plant we will never know why it grows. Likewise certain fields of influence allow growth, others do not. And other fields make it necessary to make changes.
In your own interest I would suggest to let the Greek view go and rather concentrade on a more Jewish view. The Greek view is outdated and also wrong in many ways. Heraklit was not able to grasp the deepth of the topic. Also did the available knowledge not allow him to translate his ideas into correct concepts sharp enough, but he stranded on the island he tried to build here up. The concept of wisdom goes way deeper (don't confuse it with the idea of wisdom like defined in a Greek way), but it needs to be studied extra. Lewin might be a good introduction, also he had problems too and was restricted. Then I also recommend to read the book of Proverbs and Ecclesiastes (especially chapter 3). They talk about time rather like states, comparing it to seasons.
We need a complete other language here to describe a different concept. For this we must go back to the roots and take a look at what we have missed.
Also watch this to get a feeling for the problem: https://www.youtube.com/watch?v=KSqyvvoHmAw
For an overview about the books watch this: https://www.youtube.com/watch?v=Gab04dPs_uA
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The term comes from having taken the Kantian distinction between practical (the rational in its autonomy as a priori principle of the moral law) and pragmatic (the rational as a means to an end). However, the pragmatic conception of reason is used here in a completely different context from that of Kant and strongly influenced by the theory of evolution and the concept of consciousness as a form of behavior aimed at the defense and affirmation of life. The examination is configured as a response to some kind of disturbance of a belief to which corresponded a habit, to build a new belief able to establish a new practice more appropriate and effective. So pragmaticism addressed a radical critique to the Cartesian thought, especially the criterion of truth according to which it would be possible to intuitively grasp the clear and distinct character of ideas.
The essence of pragmaticism is to recognize the operational function of thought, whereby nothing in the abstract is a ‘datum’ or a 'problem', but what that in a in a certain disturbed and indeterminate situation is a problem, which once clarified and resolved, can become a ‘datum’ in another situation and vice versa. Pragmaticism sets itself in a different position from that of empiricism as much as from that of idealism. Against empiricism, the pragmaticism denies the reduction of thought to induction or convention and states that nothing is ever given in a discreet, separate, objective manner, but that objects are events with an evident function as embedded in a link of relations that correspond to operational projects. Against the idealism of the transcendental type, pragmaticism claims instead the evolution character of thought and its link with an undetermined situation of which is the solution through the character of behaviors suitable to determine it, so there can not be any series of forms or categories ‘a priori’ rigidly defined.
Against Platonic idealism and against any ontology of an absolute character, pragmaticism argues that the 'hypostatization' of research results in eternal and immutable ideas or in absolute structures of reality, and their opposition to the actual world of experience prevent from grasping the operational character of thinking and do nothing but reproduce unwittingly a social situation of the historical and classist division of labor. If, then, with Dewey, pragmaticism proposes in the logical-epistemological field a claim of continuity of research and of its ability to self-rectify at any level, because of its experimental and instrumental nature, in ethics this claim results in a vigorous polemic against any possible ontological division ‘a priori’ between ends and means, as if were values constituted in itself and for itself and man had no possibility  of subordinating.
In general, the pragmatist thinkers believe that the search for truth or absolute certainty is a nonsense, while confer great importance to the problem of objectivity; it is, however, of an "objectivity-for-us", that is to say of a concept always linked to the "points of view", both individual and collective, from which human beings come into contact with the reality and evaluate it. If, for example, anyone reflects on the way in which we acquire our beliefs, it is easy in their view to see that the beliefs that we entertain about the way reality is structured, and therefore our ideas, perceptions and theories form an "articulated" system.
However, very often new beliefs arise that are not easily reconciled with the previous ones. In this case the scheme proposed by pragmatists to explain our behavior in the face of novelty has a markedly Darwinian character, and is based on the concept of "struggle for survival" (similar, in the end, to the "conjecture-refutation" method proposed by Karl Popper). This means that, when it emerges a new belief, we try to place it in the network of oldest believes. It is clear that if the operation succeeds, no problems arise. But things are not always so simple, and it may be the case that a new belief proves, on the one hand, highly plausible and, on the other hand, able to change the system in depth.
And that is why we are witnessing a struggle for survival between theories, beliefs and views of the world; only the most suitable survive, as they give rise to better explanations of the surrounding reality.
The two key-concepts on which rests the edifice of speculative pragmatism are "utility" and "practice." Assuming that all human beings share a set of perceptions and representations, their representatives claim that this provides an objective basis for action. Such an objectivity is however different from the classical one, understood as perspicuous representation of a reality independent of the subject. Rather, it is a "weak" objectivity, based essentially on inter-subjective criteria, which is also the only one we have. Here, then, that we can speak of theories better than others, pointing out that they are "better" only in the sense they allow us - temporarily - to organize in a more appropriate way  the sensory perceptions. The struggle between beliefs and theories of which it was said before, however, preclude to attribute the adjective "best" any character of absoluteness, because we must recognize the character perpetually contingent of our knowledge.
It follows, among other things, that (i) the meaning of a concept is "wholly" determined by the practical consequences of its application, and (ii) it must give up all "philosophia prima" to embrace a worldview devoid of metaphysical presuppositions. Although it should be noted that there are significant differences between the various representatives of the current, there is no doubt that there exists among them a basic consensus on the thesis (i) and (ii). From these pragmatists will derive also (iii) the futility of each Cartesian theory (dualistic) of knowledge, and (iv) the underlying trend towards "naturalism", namely the reduction of all knowledge to purely natural elements . Truth and objectivity are "internal" to the cognitive practices of individuals in the first place, and of the various social groups in the second place.
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Interesting question as ever Gianrocco - and of course a full analysis would be very long, I'll just try and contribute a couple of thoughts.
First, I'd be very leery of 'usefulness' as a criterion. Very many arrantly wrong 'theories' have historically been very useful in serving a specific purpose (the belief in the godhood of the emperor heightening the fighting ability of kamikazes, for instance, and so on.)
Second - very many theories can actually be modeled mathematically, although we seldom realize this, and the truth tends to lie in the numbers. To model successfully of course we need to 1- establish the suitably of math within the given domain where modeling will be used to approach the truth and 2- we need to be very good at math modeling which can be sneakily difficult (the biggest traps being overlooked variables and unrecognized implicit assumptions and/or cognitive bias - a typical issue in economic modeling, for instance), but it's worth a try. If done properly, mathematical modeling can at the very least show whether a theory is self-consistent (if not, it's very likely wrong.) 
Sometimes, math modeling shows that an answer is in principle non existent, or some variable is a non-observable - which is very useful because it suggests that perhaps the question is hiding a deeper one which should be looked at instead. 
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After generating the cif and fcf files from refinement, I exploited the "checkcif" facility available at http://checkcif.iucr.org/. The Platon program suggested the following alert:
“PLAT930_ALERT_2_B Check Twin Law ( 0  0  1)[ 1  0  4]  Estimated BASF   0.22 “
My question: what’s the meaning of “( 0  0  1)[ 1  0  4]” and how to transfer the twin law into a “3 x 3 matrix”?
Thanks a lot.
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Most of the crystallographic questions posted here suggest that a lot of people think that to do structural crystallography one just needs a diffractometer and proper software. Sorry, but crystallography is a complex subject and without attending dedicated courses one never will be able to overcome the problems are met in crystallographic research.
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Like Rowe, in his paper "The meaning of φρόνησις in EE", argues that Aristotle uses the term φρόνησις in the passages 1215b2, 1216a11 and 1216a37 as wisdom, without making it clear whether it is practical or theoretical wisdom. This would reveal a non-technical use by Aristotle in EE and would have several implications in the Aristotelian conception of ethics at this time, thus leading to the possibility of ethics still be seen as a theoretical science, approaching the EE of Platonic influence on Aristotle. 
I don't "buy" the complete Rowe's interpretation, but I have to admit that the meaning of φρόνησις in those passages is very ambiguous.
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Dear Ferenc Hörcher:
"On the whole I think Aristotle's practical philosophy makes sense only if distinguishable from his mater's one. However, I go with Gadamer, who claims that there are strong Platonic building blocks in this theory"
Of course, as soon as one wishes to determine the relationship between Aristotelian thought and Platonic philosophy, one runs immediately into two fundamental problems. The first is that each author seems to have left us diametrically opposed, or at least opposite, literature. For Aristotle, we have something akin to his lecture notes: material intended for the Academy and students. We lack the more formal composition that was "literature" vs. letters, mandates, court scripts (Antiphon's likely fictional oratory I'd count as literature, but in general prepared defenses that come down to us I wouldn't as they were not meant to be and were not considered to be).  For Plato, all we have is his literary works (and maybe a couple of letters admit the pseudepigrapha. Aristotle himself was one of the first to demarcate writings into genre and analyze the importance and nature of each category (περὶ ποιητικῆς αὐτῆς τε καὶ τῶν εἰδῶν αὐτῆς/”concerning literature both itself and the many forms it has…” Poetics 1447a). Plato not only dealt with genre but with the dangerous power literature and/or oral history/tradition had. I think that their recognition of genre as in some sense a medium both fixing and dictating the content of composition renders the distinctions between the 'kind of writing left to us by each all the more important when it comes to interpretation.
Then there's the Socratic problem. Whatever influences Plato had on Aristotle, Aristotle's works were his own. On the most superficial and inaccurate interpretation, Plato just repeated the teachings of Socrates. This is, I think, clearly wrong, but it is quite likely that Plato didn't just use Socrates as a mouthpiece, especially in his earlier works.
So we have Aristotle writing a different kind of literature for a different kind of audience, and the works of Plato which are not only literary (Aristotle was first tell us of τούς Σωκρατικούς λόγους as a genre of sorts), but also tied quite directly to what Socrates taught in ways that are impossible to wholly separate.
In short, I guess I would just agree that such a broad view of both philosophers' thought can't be of much help here. It is within the text and secondarily the corpus that we have our best chance of understanding the usage here. The problem then becomes partly a matter of the Greek language (not that Greek was or is alone here). In the Romance languages (even Latin itself to some extent) and English that we find (like the use of a numerical "alphabet" which differs from the written) a large influx and coining of technical terms using classical languages. One need look no further than German to see what a difference this can make. Greek is harder, as technically there weren't any technical words. Just technical contexts. 
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The platon.lst file gives a lot of information about the crystal geometry and data. What do they signify?
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You can additionally consult the manual for PLATON here: http://www.cryst.chem.uu.nl/platon/