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132-430: In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if there is a known probability distribution , the frequency of different outcomes over repeated events (or "trials")

264-491: A simple random sample , is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. A random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where

396-528: A boule of 400 to guide the work of the assembly. After the reforms of Cleisthenes, the Athenian Boule was expanded to 500 and was elected by lot every year. Each of Cleisthenes's 10 tribes provided 50 councilors who were at least 30 years old. The Boule's roles in public affairs included finance, maintaining the military's cavalry and fleet of ships, advising the generals , approving of newly elected magistrates, and receiving ambassadors. Most importantly,

528-422: A case since the issues in these major suits were regarded as affecting the community as a whole. Justice was rapid: a case could last no longer than one day and had to be completed by the time the sun set. Some convictions triggered an automatic penalty, but where this was not the case the two litigants each proposed a penalty for the convicted defendant and the jury chose between them in a further vote. No appeal

660-538: A chosen effect on the viewer. Nature provides examples of many kinds of pattern, including symmetries , trees and other structures with a fractal dimension, spirals , meanders , waves , foams , tilings , cracks and stripes. Symmetry is widespread in living things. Animals that move usually have bilateral or mirror symmetry as this favours movement. Plants often have radial or rotational symmetry , as do many flowers, as well as animals which are largely static as adults, such as sea anemones . Fivefold symmetry

792-716: A clinical intervention is used to reduce bias in controlled trials (e.g., randomized controlled trials ). Religion : Although not intended to be random, various forms of divination such as cleromancy see what appears to be a random event as a means for a divine being to communicate their will (see also Free will and Determinism for more). It is generally accepted that there exist three mechanisms responsible for (apparently) random behavior in systems: The many applications of randomness have led to many different methods for generating random data. These methods may vary as to how unpredictable or statistically random they are, and how quickly they can generate random numbers. Before

924-538: A cyclical fashion." Numbers like pi are also considered likely to be normal : Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times. Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases. In statistics, randomness is commonly used to create simple random samples . This allows surveys of completely random groups of people to provide realistic data that

1056-443: A debt to the city: see atimia ); for some Athenians, this amounted to permanent (and in fact inheritable) disqualification. Given the exclusive and ancestral concept of citizenship held by Greek city-states , a relatively large portion of the population took part in the government of Athens and of other radical democracies like it, compared to oligarchies and aristocracies. Some Athenian citizens were far more active than others, but

1188-489: A free gift of grain had arrived from the Egyptian king to be distributed among all citizens, many "illegitimate" citizens were removed from the registers. Citizenship applied to both individuals and their descendants. It could also be granted by the assembly and was sometimes given to large groups (e.g. Plateans in 427 BC and Samians in 405 BC). However, by the 4th century, citizenship was given only to individuals and by

1320-475: A general vote (usually by show of hands) of yes or no. Though there might be blocs of opinion, sometimes enduring, on important matters, there were no political parties, and likewise government —or opposition —as in the Westminster system did not exist. Voting was by simple majority . In the 5th century at least, there were scarcely any limits on the power exercised by the assembly. If the assembly broke

1452-576: A higher sex drive and consequentially if given free range to engage in society would be more promiscuous. With this in mind, they feared that women may engage in affairs and have sons out of wedlock which would jeopardize the Athenian system of property and inheritance between heirs as well as the citizenry of potential children if their parentage was called into question. In terms of intelligence, Athenian men believed that women were less intelligent than men and therefore, similarly to barbarians and slaves of

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1584-425: A jury pool of 6,000 in total. For particularly important public suits the jury could be increased by adding in extra allotments of 500. 1,000 and 1,500 are regularly encountered as jury sizes and on at least one occasion, the first time a new kind of case was brought to court (see graphē paranómōn ), all 6,000 members of the jury pool may have attended to one case. The cases were put by the litigants themselves in

1716-559: A major power, had at most 15,000. The non-citizen component of the population was made up of resident foreigners ( metics ) and slaves, with the latter perhaps somewhat more numerous. Around 338 BC the orator Hyperides (fragment 13) claimed that there were 150,000 slaves in Attica, but this figure is probably no more than an impression: slaves outnumbered those of citizen stock but did not swamp them. Only adult male Athenian citizens who had completed their military training as ephebes had

1848-561: A marked anti-professionalism. No judges presided over the courts, nor did anyone give legal direction to the jurors. With the random selection of jurors, it meant that these jurors were simply ordinary citizens with little to no legal training. Magistrates had only an administrative function and were laymen. Most of the annual magistracies in Athens could only be held once in a lifetime. There were no lawyers as such; litigants acted solely in their capacity as citizens. Whatever professionalism there

1980-510: A medium – air or water, making it oscillate as they pass by. Wind waves are surface waves that create the chaotic patterns of the sea. As they pass over sand, such waves create patterns of ripples; similarly, as the wind passes over sand, it creates patterns of dunes . Foams obey Plateau's laws , which require films to be smooth and continuous, and to have a constant average curvature . Foam and bubble patterns occur widely in nature, for example in radiolarians , sponge spicules , and

2112-553: A mixture of these, such as the tests by Kak, Phillips, Yuen, Hopkins, Beth and Dai, Mund, and Marsaglia and Zaman. Quantum nonlocality has been used to certify the presence of genuine or strong form of randomness in a given string of numbers. Popular perceptions of randomness are frequently mistaken, and are often based on fallacious reasoning or intuitions. This argument is, "In a random selection of numbers, since all numbers eventually appear, those that have not come up yet are 'due', and thus more likely to come up soon." This logic

2244-432: A population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, say research subjects, has the same probability of being chosen, then we can say the selection process is random. According to Ramsey theory , pure randomness (in the sense of there being no discernible pattern)

2376-416: A process called sortition . The assembly had four main functions: it made executive pronouncements (decrees, such as deciding to go to war or granting citizenship to a foreigner), elected some officials, legislated, and tried political crimes. As the system evolved, the last function was shifted to the law courts. The standard format was that of speakers making speeches for and against a position, followed by

2508-411: A roughly pyramidal form, where elements of the pattern repeat in a fractal -like way at different sizes. Mathematics is sometimes called the "Science of Pattern", in the sense of rules that can be applied wherever needed. For example, any sequence of numbers that may be modeled by a mathematical function can be considered a pattern. Mathematics can be taught as a collection of patterns. Gravity

2640-400: A special vote with a quorum of 6,000. This was generally done as a reward for some service to the state. In the course of a century, the number of citizenships so granted was in the hundreds rather than thousands. Xenias Graphe (ξενίας γραφή) was an action brought against any one who unlawfully exercised the rights of citizenship. If convicted, the person was sold as a slave and the property

2772-404: A suit, a smaller kind known as dike (δίκη) or private suit, and a larger kind known as graphe or public suit. For private suits, the minimum jury size was 200 (increased to 401 if a sum of over 1,000 drachmas was at issue), for public suits 501. Under Cleisthenes's reforms, juries were selected by lot from a panel of 600 jurors, there being 600 jurors from each of the ten tribes of Athens, making

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2904-515: A twelfth tribe in honour of the Pergamene king. The Athenians declared for Rome, and in 146 BC Athens became an autonomous civitas foederata , able to manage internal affairs. This allowed Athens to practice the forms of democracy, though Rome ensured that the constitution strengthened the city's aristocracy. Under Roman rule, the archons ranked as the highest officials. They were elected, and even foreigners such as Domitian and Hadrian held

3036-415: A way that gave each free resident of Attica a political function: Athenian citizens had the right to participate in assembly meetings. Solon sought to break away at the strong influence noble families had on the government by broadening the government's structure to include a wider range of property classes rather than just the aristocracy. His constitutional reforms included establishing four property classes:

3168-482: A weighted lottery to order teams in its draft. Mathematics : Random numbers are also employed where their use is mathematically important, such as sampling for opinion polls and for statistical sampling in quality control systems. Computational solutions for some types of problems use random numbers extensively, such as in the Monte Carlo method and in genetic algorithms . Medicine : Random allocation of

3300-618: A wider variety of viewpoints, which lowers the dangers of a small number of people holding all the power and promotes an informed voter. Philip II had led a coalition of the Greek states to war with Persia in 336 BC, but his Greek soldiers were hostages for the behavior of their states as much as allies. Alexander the Great 's relations with Athens later strained when he returned to Babylon in 324 BC; after his death, Athens and Sparta led several states to war with Macedonia and lost. This led to

3432-587: A ‘global fractal forest.’ The local ‘tree-seed’ patterns, global configuration of tree-seed locations, and overall resulting ‘global-forest’ patterns have fractal qualities. These designs span multiple mediums yet are all intended to lower occupant stress without detracting from the function and overall design of the space. In this series of studies, we first establish divergent relationships between various visual attributes, with pattern complexity, preference, and engagement ratings increasing with fractal complexity compared to ratings of refreshment and relaxation which stay

3564-473: Is a source of ubiquitous scientific patterns or patterns of observation. The sun rising and falling pattern each day results from the rotation of the earth while in orbit around the sun. Likewise, the moon's path through the sky is due to its orbit of the earth. These examples, while perhaps trivial, are examples of the "unreasonable effectiveness of mathematics" which obtain due to the differential equations whose application within physics function to describe

3696-404: Is approximated by randomization , such as selecting jurors and military draft lotteries. Games : Random numbers were first investigated in the context of gambling , and many randomizing devices, such as dice , shuffling playing cards , and roulette wheels, were first developed for use in gambling. The ability to produce random numbers fairly is vital to electronic gambling, and, as such,

3828-490: Is driven by a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment). Installations of these composite mid-high complexity ‘global-forest’ patterns consisting of ‘tree-seed’ components balance these contrasting needs, and can serve as a practical implementation of biophilic patterns in human-made environments to promote occupant wellbeing. Athenian democracy Athenian democracy developed around

3960-494: Is found in the echinoderms , including starfish , sea urchins , and sea lilies . Among non-living things, snowflakes have striking sixfold symmetry : each flake is unique, its structure recording the varying conditions during its crystallisation similarly on each of its six arms. Crystals have a highly specific set of possible crystal symmetries ; they can be cubic or octahedral , but cannot have fivefold symmetry (unlike quasicrystals ). Spiral patterns are found in

4092-410: Is impossible, especially for large structures. Mathematician Theodore Motzkin suggested that "while disorder is more probable in general, complete disorder is impossible". Misunderstanding this can lead to numerous conspiracy theories . Cristian S. Calude stated that "given the impossibility of true randomness, the effort is directed towards studying degrees of randomness". It can be proven that there

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4224-511: Is infinite hierarchy (in terms of quality or strength) of forms of randomness. In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw dice to determine fate, and this later evolved into games of chance. Most ancient cultures used various methods of divination to attempt to circumvent randomness and fate. Beyond religion and games of chance , randomness has been attested for sortition since at least ancient Athenian democracy in

4356-407: Is not a coincidence that 6,000 is the number both for the full quorum for the assembly and for the annual pool from which jurors were picked for particular trials. By the mid-4th century, however, the assembly's judicial functions were largely curtailed, though it always kept a role in the initiation of various kinds of political trial. The central events of the Athenian democracy were the meetings of

4488-559: Is only correct if applied to a system where numbers that come up are removed from the system, such as when playing cards are drawn and not returned to the deck. In this case, once a jack is removed from the deck, the next draw is less likely to be a jack and more likely to be some other card. However, if the jack is returned to the deck, and the deck is thoroughly reshuffled, a jack is as likely to be drawn as any other card. The same applies in any other process where objects are selected independently, and none are removed after each event, such as

4620-532: Is opposed to that component of its variation that is causally attributable to the source, the signal. In terms of the development of random networks, for communication randomness rests on the two simple assumptions of Paul Erdős and Alfréd Rényi , who said that there were a fixed number of nodes and this number remained fixed for the life of the network, and that all nodes were equal and linked randomly to each other. The random walk hypothesis considers that asset prices in an organized market evolve at random, in

4752-464: Is predictable. For example, when throwing two dice , the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability , and information entropy . The fields of mathematics, probability, and statistics use formal definitions of randomness, typically assuming that there

4884-458: Is random if and only if it withstands all recursively enumerable null sets. The other notions of random sequences include, among others, recursive randomness and Schnorr randomness, which are based on recursively computable martingales. It was shown by Yongge Wang that these randomness notions are generally different. Randomness occurs in numbers such as log(2) and pi . The decimal digits of pi constitute an infinite sequence and "never repeat in

5016-415: Is reflective of the population. Common methods of doing this include drawing names out of a hat or using a random digit chart (a large table of random digits). In information science, irrelevant or meaningless data is considered noise. Noise consists of numerous transient disturbances, with a statistically randomized time distribution. In communication theory , randomness in a signal is called "noise", and

5148-566: Is some 'objective' probability distribution. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space . This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences . A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions . These and other constructs are extremely useful in probability theory and

5280-651: Is speculated to have not simply been a legal obligation to the state, but also a form of ethnic-nationality. The title of "Athenian" was given to free residents deeming them citizens and granted them special privileges and protections over other residents in the city who were considered "non-citizens". In the timeline of Athenian laws, Solon's laws outlined a clear boundary between the protections that exist between citizens, Athenians, who were considered free and non-citizens, non-Athenians, who legally could be subjected to slavery. Also excluded from voting were citizens whose rights were under suspension (typically for failure to pay

5412-480: Is that a string of bits is random if and only if it is shorter than any computer program that can produce that string ( Kolmogorov randomness ), which means that random strings are those that cannot be compressed . Pioneers of this field include Andrey Kolmogorov and his student Per Martin-Löf , Ray Solomonoff , and Gregory Chaitin . For the notion of infinite sequence, mathematicians generally accept Per Martin-Löf 's semi-eponymous definition: An infinite sequence

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5544-409: Is the result of previous events, as is reflected in the concept of karma . As such, this conception is at odds with the idea of randomness, and any reconciliation between both of them would require an explanation. In some religious contexts, procedures that are commonly perceived as randomizers are used for divination. Cleromancy uses the casting of bones or dice to reveal what is seen as the will of

5676-404: Is thought that it will occur less often in the future. A number may be assumed to be blessed because it has occurred more often than others in the past, and so it is thought likely to come up more often in the future. This logic is valid only if the randomisation might be biased, for example if a die is suspected to be loaded then its failure to roll enough sixes would be evidence of that loading. If

5808-505: The Monty Hall problem , a game show scenario in which a car is hidden behind one of three doors, and two goats are hidden as booby prizes behind the others. Once the contestant has chosen a door, the host opens one of the remaining doors to reveal a goat, eliminating that door as an option. With only two doors left (one with the car, the other with another goat), the player must decide to either keep their decision, or to switch and select

5940-653: The Sicilian campaign in 413 BC, a group of citizens took steps to limit the radical democracy they thought was leading the city to ruin. Their efforts, initially conducted through constitutional channels, culminated in the establishment of an oligarchy, the Council of 400, in the Athenian coup of 411 BC . The oligarchy endured for only four months before it was replaced by a more democratic government. Democratic regimes governed until Athens surrendered to Sparta in 404 BC, when

6072-420: The agora into the assembly meeting place ( Pnyx ), with a fine being imposed on those who got the red on their clothes. After the restoration of democracy in 403 BC, pay for assembly attendance was introduced. This promoted a new enthusiasm for assembly meetings. Only the first 6,000 to arrive were admitted and paid, with the red rope now used to keep latecomers at bay. In 594 BC, Solon is said to have created

6204-473: The assembly ( ἐκκλησία , ekklesía ). Unlike a parliament , the assembly's members were not elected, but attended by right when they chose. Greek democracy created at Athens was direct , rather than representative : any adult male citizen over the age of 20 could take part, and it was a duty to do so. The officials of the democracy were in part elected by the Assembly and in large part chosen by lottery in

6336-471: The deterministic ideas of some religions, such as those where the universe is created by an omniscient deity who is aware of all past and future events. If the universe is regarded to have a purpose, then randomness can be seen as impossible. This is one of the rationales for religious opposition to evolution , which states that non-random selection is applied to the results of random genetic variation. Hindu and Buddhist philosophies state that any event

6468-454: The pentakosiomedimnoi , the hippeis , the zeugitai , and the thetes . The classifications were based on how many medimnoi a man's estate made per year with the pentakosiomedimnoi making at least 500 medimnoi, the hippeis making 300-500 medimnoi, the zeugitai making 200-300 medimnoi, and the thetes making under 200 medimnoi. By granting the formerly aristocratic role to every free citizen of Athens who owned property, Solon reshaped

6600-434: The 5th century BC, there were 10 fixed assembly meetings per year, one in each of the ten state months , with other meetings called as needed. In the following century, the meetings were set to forty a year, with four in each state month. One of these was now called the main meeting, kyria ekklesia . Additional meetings might still be called, especially as up until 355 BC there were still political trials that were conducted in

6732-469: The 6th century BC in the Greek city-state (known as a polis ) of Athens , comprising the city of Athens and the surrounding territory of Attica . Although Athens is the most familiar of the democratic city-states in ancient Greece, it was not the only one, nor was it the first; multiple other city-states adopted similar democratic constitutions before Athens. By the late 4th century BC, as many as half of

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6864-468: The Athenians provides a run-down of the structure of Athens' government and its processes. There were three political bodies where citizens gathered in numbers running into the hundreds or thousands. These are the assembly (in some cases with a quorum of 6,000), the council of 500 ( boule ), and the courts (a minimum of 200 people, on some occasions up to 6,000). Of these three bodies, the assembly and

6996-604: The Athenians their lives and did not sell them into slavery; he also restored the previous government, in 86 BC. After Rome became an Empire under Augustus , the nominal independence of Athens dissolved and its government converged to the normal type for a Roman municipality, with a Senate of decuriones . Estimates of the population of ancient Athens vary. During the 4th century BC, there might well have been some 250,000–300,000 people in Attica. Citizen families could have amounted to 100,000 people, and out of these some 30,000 would have been adult male citizens entitled to vote in

7128-628: The Boule would draft probouleumata , or deliberations for the Ecclesia to discuss and approve on. During emergencies, the Ecclesia would also grant special temporary powers to the Boule. Cleisthenes restricted the Boule's membership to those of zeugitai status (and above), presumably because these classes' financial interests gave them an incentive towards effective governance. A member had to be approved by his deme, each of which would have an incentive to select those with experience in local politics and

7260-588: The Hellenistic control of Athens, with the Macedonian king appointing a local agent as political governor in Athens. However, the governors, like Demetrius of Phalerum , appointed by Cassander , kept some of the traditional institutions in formal existence, although the Athenian public would consider them to be nothing more than Macedonian puppet dictators. Once Demetrius Poliorcetes ended Cassander's rule over Athens, Demetrius of Phalerum went into exile and

7392-462: The Peloponnesian war and there it stayed; the original amount is not known. Notably, this was introduced more than fifty years before payment for attendance at assembly meetings. Running the courts was one of the major expenses of the Athenian state and there were moments of financial crisis in the 4th century when the courts, at least for private suits, had to be suspended. The system showed

7524-406: The advent of computational random number generators , generating large amounts of sufficiently random numbers (which is important in statistics) required a lot of work. Results would sometimes be collected and distributed as random number tables . There are many practical measures of randomness for a binary sequence. These include measures based on frequency, discrete transforms , complexity , or

7656-460: The animals' appearance changing imperceptibly as Turing predicted. In visual art, pattern consists in regularity which in some way "organizes surfaces or structures in a consistent, regular manner." At its simplest, a pattern in art may be a geometric or other repeating shape in a painting , drawing , tapestry , ceramic tiling or carpet , but a pattern need not necessarily repeat exactly as long as it provides some form or organizing "skeleton" in

7788-603: The artwork. In mathematics, a tessellation is the tiling of a plane using one or more geometric shapes (which mathematicians call tiles), with no overlaps and no gaps. In architecture, motifs are repeated in various ways to form patterns. Most simply, structures such as windows can be repeated horizontally and vertically (see leading picture). Architects can use and repeat decorative and structural elements such as columns , pediments , and lintels . Repetitions need not be identical; for example, temples in South India have

7920-434: The assembly, rather than in court. The assembly meetings did not occur at fixed intervals, as they had to avoid clashing with the annual festivals that followed the lunar calendar. There was also a tendency for the four meetings to be aggregated toward the end of each state month. Attendance at the assembly was not always voluntary. In the 5th century, public slaves forming a cordon with a red-stained rope herded citizens from

8052-552: The assembly. In the mid-5th century, the number of adult male citizens was perhaps as high as 60,000, but this number fell precipitously during the Peloponnesian War. This slump was permanent, due to the introduction of a stricter definition of citizen described below. From a modern perspective, these figures may seem small, but among Greek city-states Athens was huge: most of the thousand or so Greek cities could only muster 1,000–1,500 adult male citizens each; and Corinth ,

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8184-412: The beginning of a scenario, one might calculate the probability of a certain event. However, as soon as one gains more information about the scenario, one may need to re-calculate the probability accordingly. For example, when being told that a woman has two children, one might be interested in knowing if either of them is a girl, and if yes, the probability that the other child is also a girl. Considering

8316-492: The best deterministic methods. Many scientific fields are concerned with randomness: In the 19th century, scientists used the idea of random motions of molecules in the development of statistical mechanics to explain phenomena in thermodynamics and the properties of gases . According to several standard interpretations of quantum mechanics , microscopic phenomena are objectively random. That is, in an experiment that controls all causally relevant parameters, some aspects of

8448-576: The body plans of animals including molluscs such as the nautilus , and in the phyllotaxis of many plants, both of leaves spiralling around stems, and in the multiple spirals found in flowerheads such as the sunflower and fruit structures like the pineapple . Chaos theory predicts that while the laws of physics are deterministic , there are events and patterns in nature that never exactly repeat because extremely small differences in starting conditions can lead to widely differing outcomes. The patterns in nature tend to be static due to dissipation on

8580-494: The boule was responsible for a great portion of the administration of the state, but was granted relatively little latitude for initiative; the boule's control over policy was executed in its probouleutic, rather than its executive function; in the former, it prepared measures for deliberation by the assembly, in the latter, it merely executed the wishes of the assembly. Athens had an elaborate legal system centered on full citizen rights (see atimia ). The age limit of 30 or older,

8712-497: The children is female, this rules out the boy-boy scenario, leaving only three ways of having the two children: boy-girl, girl-boy, girl-girl. From this, it can be seen only ⅓ of these scenarios would have the other child also be a girl (see Boy or girl paradox for more). In general, by using a probability space, one is less likely to miss out on possible scenarios, or to neglect the importance of new information. This technique can be used to provide insights in other situations such as

8844-628: The concept is isonomia , which is indicated also in Herodotus' discussion. Around 460 BC an individual is known with the name of Democrates , a name possibly coined as a gesture of democratic loyalty; the name can also be found in Aeolian Temnus . Athens was not the only polis in Ancient Greece that instituted a democratic resume. Aristotle points out other cities that adopted democratic governments. Most general accounts of

8976-427: The context of gambling , but later in connection with physics. Statistics is used to infer an underlying probability distribution of a collection of empirical observations. For the purposes of simulation , it is necessary to have a large supply of random numbers —or means to generate them on demand. Algorithmic information theory studies, among other topics, what constitutes a random sequence . The central idea

9108-550: The council of the Areopagus , made up of ex-archons. The members of these institutions were generally aristocrats. In 621 BC, Draco replaced the prevailing system of oral law by a written code to be enforced only by a court of law . While the laws, later come to be known as the Draconian Constitution , were largely harsh and restrictive, with nearly all of them later being repealed, the written legal code

9240-465: The courts were the true sites of power – although courts, unlike the assembly, were never simply called the demos ('the people'), as they were manned by just those citizens over thirty. Crucially, citizens voting in both were not subject to review and prosecution, as were council members and all other officeholders. In the 5th century BC, there is often a record of the assembly sitting as a court of judgment itself for trials of political importance and it

9372-420: The democracy was restored in 307 BC. However, by now Athens had become "politically impotent". An example of this was that, in 307, in order to curry favour with Macedonia and Egypt, three new tribes were created, two in honour of the Macedonian king and his son, and the other in honour of the Egyptian king. However, when Rome fought Macedonia in 200, the Athenians abolished the first two new tribes and created

9504-477: The development of Athenian democracy. Cleisthenes broke up the unlimited power of the nobility by organizing citizens into ten groups based on where they lived, rather than on their wealth. The longest-lasting democratic leader was Pericles . After his death, Athenian democracy was twice briefly interrupted by oligarchic revolutions in 411 and 404 BC, towards the end of the Peloponnesian War . It

9636-415: The die is known to be fair, then previous rolls can give no indication of future events. In nature, events rarely occur with a frequency that is known a priori , so observing outcomes to determine which events are more probable makes sense. However, it is fallacious to apply this logic to systems designed and known to make all outcomes equally likely, such as shuffled cards, dice, and roulette wheels. In

9768-557: The domination of the aristocratic families and connected every Athenian to the city's rule. Cleisthenes formally identified free inhabitants of Attica as citizens of Athens, which gave them power and a role in a sense of civic solidarity. He did this by making the traditional tribes politically irrelevant and instituting ten new tribes, each made up of about three trittyes (geographical divisions), each consisting of several demes (further subdivisions). Every male citizen over 18 had to be registered in his deme. The third set of reforms

9900-412: The elements dêmos ( δῆμος , traditionally interpreted "people" or "towns" ) and krátos ( κράτος , which means "force" or "power"), and thus means literally "people power". In the words "monarchy" and "oligarchy", the second element comes from archē ( ἀρχή ), meaning "beginning (that which comes first)", and hence also "first place or power", "sovereignty". One might expect, by analogy, that

10032-412: The emergence process, but when there is interplay between injection of energy and dissipation there can arise a complex dynamic. Many natural patterns are shaped by this complexity, including vortex streets , other effects of turbulent flow such as meanders in rivers. or nonlinear interaction of the system Waves are disturbances that carry energy as they move. Mechanical waves propagate through

10164-456: The form of a kleroterion . The formalization of odds and chance was perhaps earliest done by the Chinese of 3,000 years ago. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the 16th century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of calculus had a positive impact on

10296-401: The form of an exchange of single speeches timed by a water clock or clepsydra , first prosecutor then defendant. In a public suit the litigants each had three hours to speak, much less in private suits (though here it was in proportion to the amount of money at stake). Decisions were made by voting without any time set aside for deliberation. Jurors did talk informally amongst themselves during

10428-461: The formal study of randomness. In the 1888 edition of his book The Logic of Chance , John Venn wrote a chapter on The conception of randomness that included his view of the randomness of the digits of pi (π), by using them to construct a random walk in two dimensions. The early part of the 20th century saw a rapid growth in the formal analysis of randomness, as various approaches to the mathematical foundations of probability were introduced. In

10560-551: The gods. In most of its mathematical, political, social and religious uses, randomness is used for its innate "fairness" and lack of bias. Politics : Athenian democracy was based on the concept of isonomia (equality of political rights), and used complex allotment machines to ensure that the positions on the ruling committees that ran Athens were fairly allocated. Allotment is now restricted to selecting jurors in Anglo-Saxon legal systems, and in situations where "fairness"

10692-428: The government was placed in the hands of the so-called Thirty Tyrants , who were pro-Spartan oligarchs. After a year, pro-democracy elements regained control, and democratic forms persisted until the Macedonian army of Phillip II conquered Athens in 338 BC. Despite having its roots in the principles of classical Athens, modern democracy has developed to meet the challenges of modern administration. Direct participation

10824-484: The greatest likelihood at effective participation in government. The members from each of the ten tribes in the Boule took it in turns to act as a standing committee (the prytaneis ) of the Boule for a period of thirty-six days. All fifty members of the prytaneis on duty were housed and fed in the tholos of the Prytaneion , a building adjacent to the bouleuterion , where the boule met. A chairman for each tribe

10956-435: The impact of other visual judgments. Here we examine the aesthetic and perceptual experience of fractal ‘global-forest’ designs already installed in humanmade spaces and demonstrate how fractal pattern components are associated with positive psychological experiences that can be utilized to promote occupant wellbeing. These designs are composite fractal patterns consisting of individual fractal ‘tree-seeds’ which combine to create

11088-677: The influence of genes and the environment), and to some extent randomly. For example, the density of freckles that appear on a person's skin is controlled by genes and exposure to light; whereas the exact location of individual freckles seems random. As far as behavior is concerned, randomness is important if an animal is to behave in a way that is unpredictable to others. For instance, insects in flight tend to move about with random changes in direction, making it difficult for pursuing predators to predict their trajectories. The mathematical theory of probability arose from attempts to formulate mathematical descriptions of chance events, originally in

11220-480: The law, the only thing that might happen is that it would punish those who had proposed that it had agreed to. If a mistake had been made, from the assembly's viewpoint it could only be because it had been misled. As usual in ancient democracies, one had to physically attend a gathering to vote. Military service or simple distance prevented the exercise of citizenship. Voting was usually by show of hands (χειροτονία, kheirotonia , 'arm stretching') with officials judging

11352-426: The mathematical biologist James D. Murray and other scientists, described a mechanism that spontaneously creates spotted or striped patterns, for example in the skin of mammals or the plumage of birds: a reaction–diffusion system involving two counter-acting chemical mechanisms, one that activates and one that inhibits a development, such as of dark pigment in the skin. These spatiotemporal patterns slowly drift,

11484-486: The men. For the most part, Athens followed a citizenship-through-birth criterion. This criterion could be further divided into three categories: free birth from an Athenian father, free and legitimate birth from an Athenian father, and free and legitimate birth from an Athenian father and an Athenian mother. Athenians considered circumstances of one's birth to be relevant to the type of political identity and positions they could hold as citizens. Citizenry in ancient Athens

11616-527: The methods used to create them are usually regulated by government Gaming Control Boards . Random drawings are also used to determine lottery winners. In fact, randomness has been used for games of chance throughout history, and to select out individuals for an unwanted task in a fair way (see drawing straws ). Sports : Some sports, including American football , use coin tosses to randomly select starting conditions for games or seed tied teams for postseason play . The National Basketball Association uses

11748-491: The mid-to-late-20th century, ideas of algorithmic information theory introduced new dimensions to the field via the concept of algorithmic randomness . Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the 20th century computer scientists began to realize that the deliberate introduction of randomness into computations can be an effective tool for designing better algorithms. In some cases, such randomized algorithms even outperform

11880-443: The most general empirical patterns of the universe . Daniel Dennett 's notion of real patterns , discussed in his 1991 paper of the same name, provides an ontological framework aiming to discern the reality of patterns beyond mere human interpretation, by examining their predictive utility and the efficiency they provide in compressing information. For example, centre of gravity is a real pattern because it allows us to predict

12012-450: The movements of a bodies such as the earth around the sun, and it compresses all the information about all the particles in the sun and the earth that allows us to make those predictions. Some mathematical rule-patterns can be visualised, and among these are those that explain patterns in nature including the mathematics of symmetry, waves, meanders, and fractals. Fractals are mathematical patterns that are scale invariant. This means that

12144-515: The mutation is not entirely random however as e.g. biologically important regions may be more protected from mutations. Several authors also claim that evolution (and sometimes development) requires a specific form of randomness, namely the introduction of qualitatively new behaviors. Instead of the choice of one possibility among several pre-given ones, this randomness corresponds to the formation of new possibilities. The characteristics of an organism arise to some extent deterministically (e.g., under

12276-538: The names of wives and daughters of citizens or finding round about ways of referring to them. Pushed out of the public sphere, women's role was confined to the private sphere of working in the home and being cast as a second-rate human, subservient to her male guardian whether that be a father or husband. In the realm of Athenian men's rationalization, part of the reasons for excluding women from politics came from widely held views that women were more sexual, and intellectually handicapped. Athenian men believed that women had

12408-420: The office as a mark of honour. Four presided over the judicial administration. The council (whose numbers varied at different times from 300 to 750) was appointed by lot. It was superseded in importance by the Areopagus , which, recruited from the elected archons, had an aristocratic character and was entrusted with wide powers. From the time of Hadrian, an imperial curator superintended the finances. The shadow of

12540-508: The old constitution lingered on and archons and Areopagus survived the fall of the Roman Empire. In 88 BC, there was a revolution under the philosopher Athenion, who, as tyrant, forced the Assembly to agree to elect whomever he might ask to office. Athenion allied with Mithridates of Pontus and went to war with Rome; he was killed during the war and was replaced by Aristion . The victorious Roman general, Publius Cornelius Sulla , left

12672-998: The other door. Intuitively, one might think the player is choosing between two doors with equal probability, and that the opportunity to choose another door makes no difference. However, an analysis of the probability spaces would reveal that the contestant has received new information, and that changing to the other door would increase their chances of winning. Pattern Any of the senses may directly observe patterns. Conversely, abstract patterns in science , mathematics , or language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature and in art. Visual patterns in nature are often chaotic , rarely exactly repeating, and often involve fractals . Natural patterns include spirals , meanders , waves , foams , tilings , cracks , and those created by symmetries of rotation and reflection . Patterns have an underlying mathematical structure; indeed, mathematics can be seen as

12804-412: The outcome by sight. This could cause problems when it becomes too dark to see properly. However, any member could demand that officials issue a recount. For a small category of votes, a quorum of 6,000 was required, principally grants of citizenship, and here small colored stones were used, white for yes and black for no. At the end of the session, each voter tossed one of these into a large clay jar which

12936-450: The outcome still vary randomly. For example, if a single unstable atom is placed in a controlled environment, it cannot be predicted how long it will take for the atom to decay—only the probability of decay in a given time. Thus, quantum mechanics does not specify the outcome of individual experiments, but only the probabilities. Hidden variable theories reject the view that nature contains irreducible randomness: such theories posit that in

13068-435: The over one thousand existing Greek cities might have been democracies. Athens practiced a political system of legislation and executive bills. Participation was open to adult, free male citizens (i.e., not a metic , woman or slave.) Adult male citizens probably constituted no more than 30 percent of the total adult population. Solon (in 594 BC), Cleisthenes (in 508–07 BC), and Ephialtes (in 462 BC) contributed to

13200-421: The people. Unlike office holders (magistrates), who could be impeached and prosecuted for misconduct, the jurors could not be censured, for they, in effect, were the people and no authority could be higher than that. A corollary of this was that, at least acclaimed by defendants, if a court had made an unjust decision, it must have been because it had been misled by a litigant. Essentially there were two grades of

13332-507: The processes that appear random, properties with a certain statistical distribution are at work behind the scenes, determining the outcome in each case. The modern evolutionary synthesis ascribes the observed diversity of life to random genetic mutations followed by natural selection . The latter retains some random mutations in the gene pool due to the systematically improved chance for survival and reproduction that those mutated genes confer on individuals who possess them. The location of

13464-494: The right to practice religion. Throughout its history, Athens had many different constitutions under its different leaders. Some of the history of Athens' reforms as well as a collection of constitutions from other Ancient Greek city-states was compiled and synthesized into a large all-encompassing constitution created by either Aristotle or one of his students called the Constitution of the Athenians . The Constitution of

13596-507: The right to vote in Athens. The percentage of the population that actually participated in the government was 10% to 20% of the total number of inhabitants, but this varied from the fifth to the fourth century BC. This excluded a majority of the population: slaves , freed slaves, children, women and metics (foreign residents in Athens). The women had limited rights and privileges, had restricted movement in public, and were very segregated from

13728-435: The rise of democratic institutions refer mainly to Athens, since this Greek city-state's system provides a fuller historical record, including having the only complete extant "constitution" as compiled by Aristotle and his students in the 4th century BCE - part of a collection called Constitutions ( Politeiai ) . Before the first attempt at democratic government, Athens was ruled by a series of archons , or magistrates, and

13860-404: The roll of a die, a coin toss, or most lottery number selection schemes. Truly random processes such as these do not have memory, which makes it impossible for past outcomes to affect future outcomes. In fact, there is no finite number of trials that can guarantee a success. In a random sequence of numbers, a number may be said to be cursed because it has come up less often in the past, and so it

13992-401: The same as that for office holders but ten years older than that required for participation in the assembly, gave the courts a certain standing in relation to the assembly. Jurors were required to be under oath, which was not required for attendance at the assembly. The authority exercised by the courts had the same basis as that of the assembly: both were regarded as expressing the direct will of

14124-407: The same or decrease with complexity. Subsequently, we determine that the local constituent fractal (‘tree-seed’) patterns contribute to the perception of the overall fractal design, and address how to balance aesthetic and psychological effects (such as individual experiences of perceived engagement and relaxation) in fractal design installations. This set of studies demonstrates that fractal preference

14256-483: The search for regularities, and the output of any function is a mathematical pattern. Similarly in the sciences, theories explain and predict regularities in the world. In many areas of the decorative arts , from ceramics and textiles to wallpaper , "pattern" is used for an ornamental design that is manufactured, perhaps for many different shapes of object. In art and architecture, decorations or visual motifs may be combined and repeated to form patterns designed to have

14388-497: The sense that the expected value of their change is zero but the actual value may turn out to be positive or negative. More generally, asset prices are influenced by a variety of unpredictable events in the general economic environment. Random selection can be an official method to resolve tied elections in some jurisdictions. Its use in politics originates long ago. Many offices in ancient Athens were chosen by lot instead of modern voting. Randomness can be seen as conflicting with

14520-519: The shape of the pattern does not depend on how closely you look at it. Self-similarity is found in fractals. Examples of natural fractals are coast lines and tree shapes, which repeat their shape regardless of what magnification you view at. While self-similar patterns can appear indefinitely complex, the rules needed to describe or produce their formation can be simple (e.g. Lindenmayer systems describing tree shapes). In pattern theory , devised by Ulf Grenander , mathematicians attempt to describe

14652-425: The skeletons of silicoflagellates and sea urchins . Cracks form in materials to relieve stress: with 120 degree joints in elastic materials, but at 90 degrees in inelastic materials. Thus the pattern of cracks indicates whether the material is elastic or not. Cracking patterns are widespread in nature, for example in rocks, mud, tree bark and the glazes of old paintings and ceramics. Alan Turing , and later

14784-429: The social framework of the city-state. Under these reforms, the boule (a council of 400 members, with 100 citizens from each of Athens's four tribes) ran daily affairs and set the political agenda. The Areopagus, which formerly took on this role, remained but thereafter carried on the role of "guardianship of the laws". Another major contribution to democracy was Solon's setting up of an Ecclesia or Assembly, which

14916-458: The suppliant daughters of Danaos to take refuge in the city-state of Argos), i.e. that authority as implemented by the people in the Assembly has veto power over a king. The word is then attested in the works of Herodotus ( Histories 6.43.3) in both a verbal and nominal sense with the terms dēmokrateomai ( δημοκρατέομαι ) meaning "I participate in a democracy" and dēmokratia ( δημοκρατία ) meaning "democracy." The earlier word used for

15048-495: The term "demarchy" would have been adopted for the new form of government introduced by Athenian democrats. However, the word "demarchy" ( δημαρχία ) had already been taken and meant " mayoralty ", the office or rank of a high municipal magistrate . (In present-day use, the term " demarchy " has acquired a new meaning.) It is unknown whether the word "democracy" was in existence when systems that came to be called democratic were first instituted. The first conceptual articulation of

15180-553: The term Athenian was largely reserved for just male citizens. Before Pericles' law that decreed citizenship to be restricted to children of both Athenian men and women, the polis did not register women as citizens or keep any form of registration for them which resulted in many court cases of witnesses having to prove that women were wives of Athenian men. In addition to being barred from any form of formal participation in government, women were also largely left out of public discussions and speeches with orators going as far as leaving out

15312-544: The term is generally accepted to be c. 470 The Suppliants (l. 604) with the line sung by the Chorus: dēmou kratousa cheir ( δήμου κρατούσα χειρ ). This approximately translates as the "people's hand of power", and in the context of the play it acts as a representation of the popular requirement that the king needed to get approval from the Demos in a public assembly before taking any large decision (in this case of allowing

15444-436: The time, were considered to be incapable of effectively participating and contributing to public discourse on political issues and affairs. These rationales, as well as the barring women from fighting in battle, another requirement of citizens, meant that in the eyes of Athenian men, by nature, women were not meant to be allowed citizenship. Despite being barred from the right to vote and citizenship overall, women were granted

15576-426: The two events independently, one might expect that the probability that the other child is female is ½ (50%), but by building a probability space illustrating all possible outcomes, one would notice that the probability is actually only ⅓ (33%). To be sure, the probability space does illustrate four ways of having these two children: boy-boy, girl-boy, boy-girl, and girl-girl. But once it is known that at least one of

15708-485: The various applications of randomness . Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods , which rely on random input (such as from random number generators or pseudorandom number generators ), are important techniques in science, particularly in the field of computational science . By analogy, quasi-Monte Carlo methods use quasi-random number generators . Random selection, when narrowly associated with

15840-413: The vast numbers required for the system to work testify to a breadth of direct participation among those eligible that greatly surpassed any present-day democracy. Athenian citizens had to be descended from citizens; after the reforms of Pericles and Cimon in 450 BC, only those descended from two Athenian parents could claim citizenship. Although the legislation was not retrospective, five years later, when

15972-462: The voting procedure and juries could be rowdy, shouting out their disapproval or disbelief of things said by the litigants. This may have had some role in building a consensus. The jury could only cast a "yes" or "no" vote as to the guilt and sentence of the defendant. For private suits only the victims or their families could prosecute, while for public suits anyone ( ho boulomenos , "whoever wants to" i.e. any citizen with full citizen rights) could bring

16104-729: The world in terms of patterns. The goal is to lay out the world in a more computationally friendly manner. In the broadest sense, any regularity that can be explained by a scientific theory is a pattern. As in mathematics, science can be taught as a set of patterns. A recent study from Aesthetics and Psychological Effects of Fractal Based Design suggested that fractal patterns possess self-similar components that repeat at varying size scales. The perceptual experience of human-made environments can be impacted with inclusion of these natural patterns. Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on

16236-522: Was a hallmark of Athens' democracy, but it frequently encountered obstacles like the influence of mob mentality and hurried decision-making. Modern democratic regimes, on the other hand, place a strong emphasis on the necessity of checks and balances between the several departments of government. By ensuring that decisions are based on thoughtful discussion and careful consideration, these organizations aim to reduce rash decisions influenced by public opinion. Modern democracies have also created systems to include

16368-598: Was afterwards cracked open for the counting of the ballots. Ostracism , a unique feature of Athenian democracy introduced in the early 5th century BCE, allowed the Assembly to exile citizens deemed threats to the state's stability. This annual practice, conducted through a vote, was a preventive measure against potential tyrants and factions. As noted by Starr, ostracism exemplified Athens' efforts to safeguard democracy by placing constraints on influential figures without resorting to more severe punitive actions, thereby balancing political stability with democratic freedom. In

16500-554: Was chosen by lot each day, who was required to stay in the tholos for the next 24 hours, presiding over meetings of the Boule and Assembly. The boule also served as an executive committee for the assembly, and oversaw the activities of certain other magistrates. The boule coordinated the activities of the various boards and magistrates that carried out the administrative functions of Athens and provided from its own membership randomly selected boards of ten responsible for areas ranging from naval affairs to religious observances. Altogether,

16632-479: Was forfeited to the state. With participation in Athenian Democracy only being available to adult male Athenian citizens, women were always left out of government and public roles. Even in the case of citizenry, the term was rarely used in reference to women. Rather, women were often referred to as an astē which meant "a woman belonging to the city" or Attikē gunē which meant 'an Attic woman/wife'. Even

16764-525: Was instigated by Ephialtes in 462/1. While Ephialtes's opponents were away attempting to assist the Spartans, he persuaded the Assembly to reduce the powers of the Areopagus to a criminal court for cases of homicide and sacrilege. At the same time or soon afterward, the membership of the Areopagus was extended to the lower level of the propertied citizenship. In the wake of Athens's disastrous defeat in

16896-542: Was modified somewhat after it was restored under Eucleides ; the most detailed accounts of the system are of this fourth-century modification, rather than the Periclean system. Democracy was suppressed by the Macedonians in 322 BC. The Athenian institutions were later revived, but how close they were to the original forms of democracy is debated. The word " democracy " (Greek: dēmokratia , δημοκρατία ) combines

17028-400: Was one of the first of its kind and considered to be one of the earliest developments of Athenian democracy. In 594 BC, Solon was appointed premier archon and began issuing economic and constitutional reforms in an attempt to alleviate some of the conflict that was beginning to arise from the inequities that permeated throughout Athenian society. His reforms ultimately redefined citizenship in

17160-485: Was open to all the male citizens. Solon also made significant economic reforms including cancelling existing debts, freeing debtors, and no longer allowing borrowing on the security of one's own person as a means of restructuring enslavement and debt in Athenian society. In 561 BC, the nascent democracy was overthrown by the tyrant Peisistratos but was reinstated after the expulsion of his son, Hippias , in 510. Cleisthenes issued reforms in 508 and 507 BC that undermined

17292-401: Was possible. There was however a mechanism for prosecuting the witnesses of a successful prosecutor, which it appears could lead to the undoing of the earlier verdict. Payment for jurors was introduced around 462 BC and is ascribed to Pericles , a feature described by Aristotle as fundamental to radical democracy ( Politics 1294a37). Pay was raised from two to three obols by Cleon early in

17424-449: Was tended to disguise itself; it was possible to pay for the services of a speechwriter or logographer ( logographos ), but this may not have been advertised in court. Jurors would likely be more impressed if it seemed as though litigants were speaking for themselves. As the system evolved, the courts (that is, citizens under another guise) intruded upon the power of the assembly. Starting in 355 BC, political trials were no longer held in

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