text:
Rye

Rye (Secale cereale) is a grass grown extensively as a grain, a cover crop and a forage crop. It is grown principally in an area from eastern and northern Europe into Russia. It is much more tolerant of cold weather and poor soil than other cereals, making it useful in those regions; its vigorous growth suppresses weeds, and provides abundant forage for animals early in the year. It is a member of the wheat tribe (Triticeae) which includes the cereals wheat and barley. Rye grain is used for bread, beer, rye whiskey, and animal fodder. In Scandinavia, rye was a staple food in the Middle-ages, and rye crispbread remains a popular food in the region. Around half of world production is in Europe; relatively little is traded between countries. A wheat-rye hybrid, triticale, combines the qualities of the two parent crops and is produced in large quantities worldwide. In European folklore, the Roggenwolf ("rye wolf") is a carnivorous corn demon or Feldgeist.

Origins

The rye genus Secale is in the grass tribe Triticeae, which contains other cereals such as barley (Hordeum) and wheat (Triticum).

The generic name Secale, related to Italian segale and French seigle meaning "rye", is of unknown origin but may derive from a Balkan language. The English name rye derives from Old English ryge, related to Dutch rogge, German Roggen, Russian rozh, again all with the same meaning.

Rye is one of several cereals that grow wild in the Levant, central and eastern Turkey and adjacent areas. Evidence uncovered at the Epipalaeolithic site of Tell Abu Hureyra in the Euphrates valley of northern Syria suggests that rye was among the first cereal crops to be systematically cultivated, around 13,000 years ago. However, that claim remains controversial; critics point to inconsistencies in the radiocarbon dates, and identifications based solely on grain, rather than on chaff.

Domesticated rye occurs in small quantities at a number of Neolithic sites in Asia Minor (Anatolia, now Turkey), such as the Pre-Pottery Neolithic B Can Hasan III near Çatalhöyük, but is otherwise absent from the archaeological record until the Bronze Age of central Europe, c. 1800–1500 BCE.

It is likely that rye was brought westwards from Asia Minor as a secondary crop, meaning that it was a minor admixture in wheat as a result of Vavilovian mimicry, and was only later cultivated in its own right. Archeological evidence of this grain has been found in Roman contexts along the Rhine and the Danube and in Ireland and Britain. Roman naturalist Pliny the Elder was dismissive of a grain that may have been rye, writing that it "is a very poor food and only serves to avert starvation". He said it was mixed with spelt "to mitigate its bitter taste, and even then is most unpleasant to the stomach".

Description
Rye is a tall grass grown for its seeds; it can be an annual or a biennial. Depending on environmental conditions and variety it reaches 1 to 3 m in height. Its leaves are blue-green, long, and pointed. The seeds are carried in a curved head or spike some 7 to 15 cm long. It is composed of many spikelets, each of which holds two small flowers; the spikelets alternate left and right up the head.

Cultivation

Since the Middle Ages, people have cultivated rye widely in Central and Eastern Europe. It serves as the main bread cereal in most areas east of the France–Germany border and north of Hungary. In Southern Europe, it was cultivated on marginal lands.

Rye grows well in much poorer soils than those necessary for most cereal grains. Thus, it is an especially valuable crop in regions where the soil has sand or peat. Rye plants withstand cold better than other small grains, surviving snow cover that would kill winter wheat. Winter rye is the most popular: it is planted and begins to grow in autumn. In spring, the plants develop rapidly. This allows it to provide spring grazing, at a time when spring-planted wheat has only just germinated.

The physical properties of rye affect attributes of the final food product such as seed size, surface area, and porosity. The surface area of the seed directly correlates to the drying and heat transfer time. Smaller seeds have increased heat transfer, which leads to lower drying time. Seeds with lower porosity lose water more slowly during the process of drying.

Rye is harvested like wheat with a combine harvester, which cuts the plants, threshes and winnows the grain, and releases the straw to the field where it is later pressed into bales or left as soil amendment. The resultant grain is stored in local silos or transported to regional grain elevators and combined with other lots for storage and distant shipment. Before the era of mechanised agriculture, rye harvesting was a manual task performed with scythes or sickles.

Agroecology
Winter rye is any breed of rye planted in the autumn to provide ground cover for the winter. It grows during warmer days of the winter when sunlight temporarily warms the plant above freezing, even while there is general snow cover. It can be used as a cover crop to prevent the growth of winter-hardy weeds.

Rye grows better than any other cereal in heavy clay and light sandy soil, and infertile or drought-affected soils. It can tolerate pH between 4.5 and 8.0, but soils having pH 5.0 to 7.0 are best suited for rye cultivation. Rye grows best in fertile, well-drained loam or clay-loam soils. Rye can thrive in subzero environments, assisted by the production of antifreeze polypeptides (different from the antifreeze polypeptides produced by some fish and insects) by the leaves of winter rye.

Rye is a common, unwanted invader of winter wheat fields. If allowed to grow and mature, it may cause substantially reduced prices (docking) for harvested wheat.

Pests and diseases

The nematode Ditylenchus dipsaci and a variety of herbivorous insects can seriously affect plant health.

Rye is highly susceptible to the ergot fungus. Consumption of ergot-infected rye by humans and animals results in ergotism, which causes convulsions, miscarriage, necrosis of digits, hallucinations and death. Historically, damp northern countries that depended on rye as a staple crop were subject to periodic epidemics. Modern grain-cleaning and milling methods have practically eliminated ergotism, but it remains a risk if food safety vigilance breaks down.

After an absence of 60 years, stem rust (Puccinia graminis f. sp. tritici) has returned to Europe in the 2020s. Areas affected include Germany, Russia (Western Siberia), Spain, and Sweden.

Production and consumption

Rye is grown primarily in Eastern, Central and Northern Europe. The main rye belt stretches from northern Germany through Poland, Ukraine, and eastwards into central and northern Russia. Rye is also grown in North America, in South America including Argentina, in Oceania (Australia and New Zealand), in Turkey, and in northern China. Production levels of rye have fallen since 1992 in most of the producing nations, ; for instance, production of rye in Russia fell from 13.9 Mt in 1992 to 2.2 Mt in 2022.

World trade of rye is low compared with other grains such as wheat. The total export of rye for 2016 was $186M compared with $30.1B for wheat.

Poland consumes the most rye per person at 32.4 kg per capita (2009). Nordic and Baltic countries are also very high. The EU in general is around 5.6 kg per capita. The entire world only consumes 0.9 kg per capita.

Nutritional value
Raw rye contains 11% water, 76% carbohydrates, 10% protein, and 2% fat (table). A 100 g reference amount of rye provides 338 kcal of food energy, and is a rich source (20% or more of the Daily Value, DV) of essential nutrients, including dietary fiber, B vitamins, such as thiamine and niacin (each at 25% DV), and several dietary minerals. Highest micronutrient contents are for manganese (130% DV) and phosphorus (27% DV) (table).

Health effects
According to Health Canada and the U.S. Food and Drug Administration, consuming at least 4 g per day of rye beta-glucan or 0.65 g per serving of soluble fiber can lower levels of blood cholesterol, a risk factor for cardiovascular diseases.

Eating whole-grain rye, as well as other high-fibre grains, improves regulation of blood sugar (i.e., reduces blood glucose response to a meal). Consuming breakfast cereals containing rye over weeks to months also improved cholesterol levels and glucose regulation.

Health concerns
Like wheat, barley, and their hybrids and derivatives, rye contains glutens and related prolamines, which makes it an unsuitable grain for consumption by people with gluten-related disorders, such as celiac disease, non-celiac gluten sensitivity, and wheat allergy, among others. Nevertheless, some wheat allergy patients can tolerate rye or barley.

Food and drink
Rye grain is refined into a flour high in gliadin but low in glutenin and rich in soluble fiber. Alkylresorcinols are phenolic lipids present in high amounts in the bran layer (e.g. pericarp, testa and aleurone layers) of wheat and rye (0.1–0.3% of dry weight). Rye bread, including pumpernickel, is made using rye flour and is a widely eaten food in Northern and Eastern Europe. In Scandinavia, rye is widely used to make crispbread (Knäckebröd); in the Middle Ages it was a staple food in the region, and it remains popular in the 21st century.

Rye grain is used to make alcoholic drinks, such as rye whiskey and rye beer. The traditional cloudy and sweet-sour low-alcohol beverage kvass is fermented from rye bread or rye flour and malt.

Other uses
Rye is a useful forage crop in cool climates; it grows vigorously and provides plentiful fodder for grazing animals, or green manure to improve the soil. It forms a good cover crop in winter with its rapid growth and deep roots.

Rye straw is used as livestock bedding, despite the risk of ergot poisoning. It is used on a small scale to make crafts such as corn dollies. More recently it has found uses as a raw material for bioconversion to products such as the sweetener xylitol.

Rye flour is mixed with linseed oil and iron oxide to make traditional Falun red paint, widely used as a house paint in Sweden.

Production of hybrids

Plant breeders, starting in the 19th century in Germany and Scotland, but mainly from the 1950s, worked to develop a hybrid cereal with the best qualities of wheat and rye, now called triticale. Modern triticales are hexaploid with six sets of chromosomes, and are used to produce millions of tons of cereal annually.

Varieties of rye hold much genetic diversity, which can be used to improve other crops such as wheat. For example, the pollination abilities of wheat was improved by the addition of the rye chromosome 4R; this increased the size of the wheat anther and the amount of pollen. The 1R chromosome is the source of many crop disease resistance genes. Varieties such as Petkus, Insave, Amigo, and Imperial have donated 1R-originating resistance to wheat. AC Hazlet rye is a medium-sized winter rye with resistance to both lodging and shattering. Rye was the gene donor of Sr31 – a stem rust resistance gene – introgressed into wheat.

The characteristics of S. cereale have been combined with another perennial rye, S. montanum, to produce S. cereanum, which has the beneficial characteristics of each. The hybrid rye can be grown in harsh environments and on poor soil. It provides improved forage with digestible fiber and protein.

In human culture

In European folklore, the Roggenwolf ("rye wolf") is a carnivorous corn demon or Feldgeist, a field spirit shaped like a wolf. The Roggenwolf steals children and feeds on them. The last grain heads are often left at their place as a sacrifice for the agricultural spirits.

In contrast, the Roggenmuhme or Roggenmutter ("rye aunt" or "rye mother") is an anthropomorphic female corn demon with fiery fingers. Her bosoms are filled with tar, and may end in tips of igneous iron. Her bosoms are also long, and as such must be thrown over her shoulders when she runs. The Roggenmuhme is completely black or white, and in her hand she has a birch or whip from which lightning sparks. She can change herself into different animals; such as snakes, turtles, frogs and others.

The classical scholar Carl A. P. Ruck writes that the Roggenmutter was believed to go through the fields, rustling like the wind, with a pack of rye wolves running after her. They spread ergot through the sheaves of harvested rye. According to Ruck, they then lured children into the fields to nurse on the infected grains "like the iron teats of the Roggenmutter". The enlarged reddish ergot-infected grains were known as Wulfzähne (wolf teeth).

title: Rye


text:
Tare

Tare or Tares may refer to:

 * Tare (armour), a leg and groin protector used in a number of Japanese martial arts
 * Tare (legume), several plants of the legume family
 * Tare (surname), a surname
 * Tare (tufted grass), a genus of nine species of tufted grasses
 * Tare, Rwanda
 * Tare River, in Romania
 * Tare sauce, a Japanese dipping sauce
 * Tare weight, the weight of an empty vehicle or container
 * Tares, Iran
 * Vicia sativa, a plant also known as the tare
 * Parable of the Tares, a parable of Jesus, referring to tares (flowering plants) as weeds

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Tare


text:
Turnip

The turnip or white turnip (Brassica rapa subsp. rapa) is a root vegetable commonly grown in temperate climates worldwide for its white, fleshy taproot. Small, tender varieties are grown for human consumption, while larger varieties are grown as feed for livestock. The name turnip – used in many regions – may also include rutabaga, neep or swede.

Etymology
The origin of the word turnip is uncertain, though it is hypothesised that it could be a compound of turn as in turned/rounded on a lathe and neep, derived from Latin napus, the word for the plant. According to An Universal Etymological English Dictionary, turn refers to "round napus to distinguish it from the napi, which were generally long".

Description
The most common type of turnip is mostly white-skinned apart from the upper 1 to 6 cm, which protrude above the ground and are purple or red or greenish where the sun has hit. This above-ground part develops from stem tissue, but is fused with the root. The interior flesh is entirely white. The root is roughly globular, from 5 - 20 cm in diameter, and lacks side roots. Underneath, the taproot (the normal root below the swollen storage root) is thin and 10 cm or more in length; it is often trimmed off before the vegetable is sold. The leaves grow directly from the above-ground shoulder of the root, with little or no visible crown or neck (as found in rutabagas).

Turnip leaves are sometimes eaten as "turnip greens" ("turnip tops" in the UK), and they resemble mustard greens (to which they are closely related) in flavor. Turnip greens are a common side dish in southeastern U.S. cooking, primarily during late Fall and Winter. Smaller leaves are preferred. Varieties of turnip grown specifically for their leaves resemble mustard greens and have small roots. These include rapini (broccoli rabe), bok choy, and Chinese cabbage. Similar to raw cabbage or radish, turnip leaves and roots have a pungent flavor that becomes milder after cooking.

Turnip roots weigh up to 1 kg, although they are usually harvested when smaller. Size is partly a function of variety and partly a function of the length of time the turnip has grown.

Nutrition
Boiled green leaves of the turnip top ("turnip greens") provide 20 kcal of food energy in a reference serving of 100 g, and are 93% water, 4% carbohydrates, and 1% protein, with negligible fat (table). The boiled greens are a rich source (more than 20% of the Daily Value, DV) particularly of vitamin K (350% DV), with vitamin A, vitamin C, and folate also in significant content (30% DV or greater, table). Boiled turnip greens also contain substantial lutein (8440 micrograms per 100 g).

In a 100-gram reference amount, boiled turnip root supplies 22 kcal, with only vitamin C in a moderate amount (14% DV). Other micronutrients in boiled turnip are in low or negligible content (table). Boiled turnip is 94% water, 5% carbohydrates, and 1% protein, with negligible fat.

History
Wild forms of the turnip and its relatives, the mustards and radishes, are found over western Asia and Europe. Starting as early as 2000 BCE, related oilseed subspecies of Brassica rapa like oleifera may have been domesticated several times from the Mediterranean to India, though these are not the same turnips cultivated for its roots. Previous estimates of domestication dates are limited to linguistic analyses of plant names.

Edible turnips were first domesticated in Central Asia several thousand years ago, supported by genetic studies of both wild and domesticated varieties showing Central Asian varieties are the most genetically diverse crops. Ancient literary references to turnips in Central Asia, and the existence of words for 'turnip' in ancestral languages of the region, also support the turnip as the original domesticated form of Brassica rapa subsp. rapa. It later spread to Europe and East Asia with farmers in both areas later selecting for larger leaves; it subsequently became an important food in the Hellenistic and Roman world. The turnip spread to China, and reached Japan by 700 CE.

Turnips were an important crop in the cuisine of Antebellum America. They were grown for their greens as well as the roots, and could yield edible greens within a few weeks of planting, making them a staple of new plantations still in the process of becoming productive. They could be planted as late as the fall and still provide newly arrived settlers with a source of food. The typical southern way of cooking turnip greens was to boil them with a chunk of salt pork. The broth obtained from this process was known as pot likker and was served with crumbled corn pone, often made from coarse meal when little else was available along the antebellum frontier.

Cultivation
The 1881 American Household Cyclopedia advises that turnips can be grown in fields that have been harrowed and ploughed. It recommends planting in late May or June and weeding and thinning with a hoe throughout the summer.

As a root crop, turnips grow best in cool weather; hot temperatures cause the roots to become woody and bad-tasting. They are typically planted in the spring in cold-weather climates (such as the northern US and Canada) where the growing season is only 3–4 months. In temperate climates (ones with a growing season of 5–6 months), turnips may also be planted in late summer for a second fall crop. In warm-weather climates (7 or more month growing season), they are planted in the fall. 55–60 days is the average time from planting to harvest.

Turnips are a biennial plant, taking two years from germination to reproduction. The root spends the first year growing and storing nutrients, and the second year flowers, produces seeds, and dies. The flowers of the turnip are tall and yellow, with the seeds forming in pea-like pods. In areas with less than seven-month growing seasons, temperatures are too cold for the roots to survive the winter. To produce seeds, pulling the turnips and storing them over winter is necessary, taking care not to damage the leaves. During the spring, they may be set back in the ground to complete their lifecycle.

Relevance in human use
In England around 1700, Charles "Turnip" Townshend promoted the use of turnips in a four-year crop-rotation system that enabled year-round livestock feeding. In the United States, the smaller white vegetables are called turnips, while the larger yellow ones are referred to as rutabagas, a borrowing from Swedish rotabagge (a dialectal word from Västergötland). In most of England, turnips are the same, while the larger yellow ones are referred to as swedes. In Scotland, the larger yellow vegetables are referred to as turnips.

In the Austrian region of Wildschönau farmers produce a kind of schnaps called Krautinger from a variation of Brassica rapa ssp. Rapa, since they were granted permission to do so under Empress Maria Theresia in the 18th century. It is notorious for its distinct taste and smell.

Heraldry
The turnip is an old vegetable charge in heraldry. It was used by Leonhard von Keutschach, prince-archbishop of Salzburg. The turnip is still the heart shield in the arms of Keutschach am See.

The arms of the former municipality of Kiikala, Finland, were Gules, a turnip Or.

title: Turnip


text:
Talk:L-system

Tilings
This paragraph is unclear to me:

 * As an L-system these tilings are called Penrose's rhombuses and Penrose's tiles. Above pictures were generated for n=6 as an L-system. If we properly superimpose Penrose tiles as an L-system we get next tiling:

It doesn't define what n is. It doesn't define what "properly superimpose as an L-system" means. It doesn't say which L-system was used to generate those images. I can't guess these from context. --LC

in the canter dust example,it contains a line of "constant" about some 60 degrees. what's that line doing there? I think it is wrong.

in the Koch snowflake example, shouldn't the plus and minus sign be part of the alphabet?

i think the penrose tiling example is a bit forced. Penrose tilings are aperiodic. So, i doubt it can be reduced to a string replacement system.

also, i'm not sure L-Sytem has a place in serious math. If it does, it is or relates to formal logic. Should there be a mention?

thanks.

Xah P0lyglut 11:18, 2004 Jan 15 (UTC)

Constants?
In Example 3, Cantor Dust, what are the constants for? (I'm not even sure if that's one, two, or three constants you're trying to list.) You never seem to use them.

Concerning the Koch curve shouldn't + and - be constants (clearly there're no productions for them). According to the definition stated, the alphabet consists of variables, hence I disagree with the poster asking that they should be included in the alphabet.

Koch curve
I would like to see one higher order iteration (n=20 ?) of the Koch curve. IMO, the result is illustrative and pretty impressive especially for people new to the subject.

Need new Penrose time images
The three images of Penrose tilings are all untagged, and they don't have authorship info (dating from before "the big conversion"). I guess when the next untagged-image jihad gets underway these images may well meet with infinite justice, so we need to start thinking about replacements for them ASAP. It's a bummer, as they're really attractive images. -- Finlay McWalter | Talk 15:17, 7 October 2005 (UTC)

 * These images were made by me with Winfract version 18.21, and we can tag them as public domain. I'll do that right the way. --xJaM 14:02, 21 October 2005 (UTC)

Example 1 seems strange to me
Lindenmayer's original L-system for modelling the growth of algae.

 * variables :  A B
 * constants :  none
 * start    :   A
 * rules    :   (A → AB), (B → A)

which produces:

 * n=0 : A → AB
 * n=1 : AB → ABA
 * n=2 : ABA → ABAAB
 * n=3 : ABAAB → ABAABABA

I would expect it to look like this more in keeping with how the others are shown:
 * n=0 : A
 * n=1 : AB
 * n=2 : ABA
 * n=3 : ABAAB
 * n=4 : ABAABABA

Am I missing something? Hogan 02:11, 29 April 2006 (UTC)

 * Yes, you are right - the format of Example 1 was not consistent with the format of other examples. I have fixed it. Gandalf61 08:16, 2 May 2006 (UTC)

As I had problmes to follow the intial example I would suggest this (only in the first example, to make the principle more clear): n=0:        A              start/axiom/initiator / \   n=1:       A   B            the single A spawns into an A followed by a B              /|    \ n=2:    A B     A          the former A again spawns into AB, B turnes into A            /| |     |\ n=3:  A B A     A B        note all A's producing a copy of themself in the first place, then a B, which turns /| | |\   |\ \    n=4: A B A A B   A B A      into an A one generation later, starting to spawn/repeat/recurse then

which is inspired by the (existing) external link (13 MB!). Maybe an image instead of the ASCII graphic would be even more apropriate?

Also the Fibonacci example 2 refers to example 1. Then ex. 1 should have a start/axiom/initiator of B (and the above diagram extended accordingly), else the reference doesn't hold true! - Deerwood (talk) 05:02, 1 July 2008 (UTC)

Example 1 seems clearly wrong. Also, it's unclear what the terminology (A → AB), (B → A) means. Does (A → AB) mean "A gets followed by AB"? Or does it mean "A gets replaced by AB"?

If (A → AB means "A gets followed by AB" then example 1 is clearly incorrect. In that case, we would have

 * n=0 : A
 * n=1 : AAB
 * n=2 : AABA
 * n=3 : AABAAB
 * n=4 : AABAABA and so on. In short, you'd always get nothing but a repetition of AAB over and over again. Since that's obviously not what an L-system does, your explanation doesn't seem to have made clear what's going on.

If, on the other hand, (A → AB) means "A gets replaced by AB," then your first example is also clearly incorrect. In that case, we would get

 * n=0 : A
 * n=1 : AB
 * n=2 : ABA
 * n=3 : ABAB
 * n=4 : ABABA and so on. In short, a mere repetition of AB over and over again. This is also clearly not what an L-system does.

So either I'm completely missing something, or the rules you've given are being used in some way that's not at all clear from your explanation, or Example 1 is simply incorrect.

Either way, some more clarification of exactly what the symbols mean and precisely how the rules are supposed to operate would be a big help. — Preceding unsigned comment added by 140.211.8.7 (talk) 20:28, 31 October 2013 (UTC)

Another example: Heighway dragon
Another nice example is a dragon curve. It is described by the following L-system:
 * variables :  L R
 * constants :  + −
 * start    :   R
 * rules    :   (R → R+L), (L → R−L)

meaning, respectively:
 * L, R — step forward
 * plus — turn right by 90°
 * minus — turn left by 90°

Some initial strigns are:
 * n=0 : R
 * n=1 : R+L
 * n=2 : R+L+R−L
 * n=3 : R+L+R−L+R+L−R−L
 * n=4 : R+L+R−L+R+L−R−L+R+L+R−L−R+L−R−L

CiaPan 20:53, 6 October 2006 (UTC)

Merge with graftal article?
I've added graftal to the "See Also" section, but does it really need a separate article? There doesn't seem to be a whole lot of difference between L-systems and graftals. Perhaps the articles should be merged together. What do you think? -- Sakurambo 桜ん坊  13:53, 15 May 2007 (UTC)

This is a good suggestion, since, as you noted, there is not much difference. In fact, a graftal is an L-System. So I definitely support this idea. Kwvan (talk) 18:29, 12 October 2009 (UTC)

Better examples
Some of the examples given in this article are a bit rubbish. The first example provides no explanation of what is going on. What does the string "ABAABABAABAABABAABABAABAABABAABAAB" have to do with algae? If the only important information is the length of each sequence, then how does this differ from Fibonacci's modelling of rabbit populations? The second example is completely redundant — the first example also generates a Fibonacci sequence, in case you hadn't noticed, and in any case, there's a much simpler algorithm for calculating this series, so in what way is this a useful example of the capabilities of an L-system? Examples 5 and 9 are quite useless without the rules used to produce these figures. -- Sakurambo 桜ん坊  14:10, 15 May 2007 (UTC)

 * The Penrose tiling algorithm is given at the article of the same name, though the syntax is different. I could try translating between notations myself, but I am loathe to introduce errors. The modified Koch curve I'm not sure about, from the Koch curve page the standard curve uses

Alphabet : F   Constants : +, − Axiom : F++F++F Production rules: F → F−F++F−F I am guessing that the variant would alternate the production rule between F → F−F++F−F and F → F+F--F+F though this is pure speculation from the description, I have no idea how it was actually generated. As to the criticism of the first examples, algae was merely what he was historically trying to model the growth of, I don't think it is meant to represent the useful capabilities of the system, more to provide an example that correlates it with a well known algorithm. Nazlfrag (talk) 06:27, 13 June 2008 (UTC)

Explanations
I added an explanation ASCII art to the first example and hope this doesn't get deleted immediately without discussion.

Visitors/readers of an encyclopedia ain't expected to be mathematicians, programmers or specialists in any way, are they? Readers should at least be able to understand the concept/essentials ... may be, one or the other reader then gets inspired to read on, learn and understand? And, may be, contribute? - Deerwood (talk) 03:38, 3 July 2008 (UTC)

IFS
Is there a reference for the statement that "L-systems can also be used to generate self-similar fractals such as iterated function systems"? Richard Pinch (talk) 21:17, 16 July 2008 (UTC)

I removed that bit about IFSs in the intro, as I thought it was stated in a misleading way. In my understanding, IFSs and L-systems are considered two different methods both able to generate fractal objects. Oftentimes an L-system (usually with Turtle interpretation) and an IFS can generate the same object (such as the Koch curve) however in these cases we wouldn't say the L-system is generating an IFS itself (i.e., the set of transformations). The formal relationship between IFSs and L-systems is an area of research which could serve as its own section. There are methods of constructing equivalent IFSs out of certain kinds of L-systems (equivalent meaning they each generate the same object), and also methods of expressing certain kinds of IFSs as equivalent L-systems. Because they are quite expressive, perhaps you *could* have an L-system which quite literally generates an IFS. However, I don't think that's what the intro was getting at, as L-system fractal generation is most often written about in the context of Turtle graphics. Druggiero (talk) 02:08, 18 March 2018 (UTC)

Examples...
Example 1 and 2 are practically the same system; having both is somewhat redundant.

Example 3(?) the dust example, should be changed to use conventional notation namely F for draw forward and f for move forward.

Example 5(?) the penrose tilings example, doesn't supply the rules for the system; the rules need to be added or the example scrapped.

All the examples really need reworking; the term 'constants' isn't one that's really used in any of the published papers on L-systems.

The article could use an example of the various common types of L-system, stochastic (random), parametric and context sensitive.

87.194.144.173 (talk) 15:36, 3 May 2010 (UTC)

Cantor example is nothing like the diagram
The rules give: n = 1 : ABA n = 2 : ABABBBABA n = 3 : ABABBBABABBBBBBBBBABABBBABA n = 4 : ABABBBABABBBBBBBBBABABBBABABBBBBBBBBBBBBBBBBBBBBBBBBBBABABBBABABBBBBBBBBABABBBABA now only if each line is scaled to the same lenghth might you get something like the picture. --Paddy (talk) 23:33, 14 May 2011 (UTC)

 * The scaling is so typical, perhaps it was overseen to mention it. The L-system itself is just the evolution of the character strings from one generation to the next. On top of that one may feed the character string of each generation as command sequence into a drawing machine like the (LOGO) Turtle. To obtain convergence of some kind one has to care how the characters relate to commands. This means adjusting the angles and step sizes, and also the orientation of the turtle at the start point, if in 2D. In the Cantor example this is simple -- three times as many characters, each a move forward, means one third of the step size from one generation to the next.--LutzL (talk) 13:19, 15 May 2011 (UTC)

Chomskys "Ultra-conservative" views
Without wanting to make any statement about the facts, could the sentences "yes: in language theory, Chomsky is ultra-conservative" please be reworked or removed? This does not sound very encyclopedia-like. In general, the whole paragraph "Chomsky's conservative...refer to Chomsky.[2]" should probably not in the introduction of the article but maybe at some later point. 95.117.217.100 (talk) 11:24, 25 July 2011 (UTC)

 * I have removed that paragraph from the article's lead. I am not completely sure what the editor who wrote this was trying to say, but in any case it has no relevance to the subject of this article. Gandalf61 (talk) 11:40, 25 July 2011 (UTC)

Context-free systems
I don't know L-systems so well but in general formal languages the context-free ones are a strictly larger class that the mentioned equivalents of regular languages. It seems to apply here as well. Can someone competent look at it and eventually fix it? Neználek (talk) 13:17, 11 April 2012 (UTC)

 * See the "subset or superset" talk topic below. 67.198.37.16 (talk) 20:53, 11 February 2024 (UTC)

This Article Needs A ReWrite
As it is now, I can't tell if it's supposed to be about some type of mathematics or what. The article has tons of images of random plants and is talking all about biology and other non-math things. I'm pretty sure the concept of "L-System" is purely mathematical.... please remove these ancillary or tangential "connections" or at least dont make them the focus of the article anymore. — Preceding unsigned comment added by 71.201.95.139 (talk) 21:02, 8 May 2013 (UTC)

 * L-systems are a mathematical concept that have applications in modelling plants and other organisms. We can't write about L-systems without including these applications. However, I have modified the lead paragraph so that it says what an L-system is before it mentions applications in biology. Gandalf61 (talk) 08:48, 9 May 2013 (UTC)

Edit by RCB I think so too, the article is not precise. Like for example "growth patterns of various types of algae, such as the blue/green bacteria Anabaena catenula." - Algae are NOT BACTERIA! — Preceding unsigned comment added by 62.61.159.141 (talk) 13:38, 22 June 2015 (UTC)
 * The thing the article is talking about is what is known as Cyanobacteria. One of the common names for cyanobacteria is "blue-green algae". It may be a misnomer, but it is not at odds with common usage. — Preceding unsigned comment added by 2003:69:CD6A:F401:D189:4DF8:E1B0:B7D0 (talk) 11:53, 15 September 2015 (UTC)
 * It isn't at odds with common use of blue-green algae, but the term algae does not include blue-green algae. Referring to the one as the general case, and the other as a specific case, is plainly wrong. The Real Marauder (talk) 21:18, 18 May 2017 (UTC)

Subset or Superset of Languages
The article states in the "L-system structure" section that "L-systems are strict subsets of languages". Since formal languages are produced by applying only one production rule at a time in each iteration, but L-systems apply all possible matching production rules in each iteration, shouldn't L-systems be considered supersets of languages (or, equivalently, that languages are strict subsets of L-systems)? At present, there is no citation or reference for the subset claim. — Loadmaster (talk) 15:40, 25 October 2016 (UTC)

 * "Languages", as generally conceived, consist of a collection of production rules (or term rewriting rules, or syntax - syntactical grammar elements) applied at one fixed location in space (usually a nebulous location, such as "inside the computer"). The conventional Chomsky hierarchy applies only to this concept of a language happening only at one location. By contrast, L-systems happen in many physical locations, all at the same time. The idea is that each biological cell contains DNA/RNA that encodes a term rewriting system or some syntax/grammar, and, as that cell, as it lives and grows, expresses that grammar, at the location where that cell is located. This is related to the idea of a cellular automaton as well as to distributed computing.

 * So far, so good. Now here is the trick: you can take a pencil, and draw a big box around the entire L-system, and say "everything inside of there is happening at one location", and define a "big" language that is the (Cartesian) product of all the "small" individual languages running in each cell. The general theorem is that a Cartesian product of multiple Turing machines (or of finite automata) is no more powerful than a single Turing machine (or finite automaton). Thus, in the conventional Chomsky hierarchy, they are at the same level, the same power. The resulting language is neither a superset nor a subset, but equally expressive.

 * The above holds true when one is working with a fixed, finite number of automata/turing machines (a finite-dimensional Cartesian product). The only way to evade this conclusion about the expressive power of languages is to take the $$n\to\infty$$ limit. In this limit things ... work differently. The result of that limit is called a geometric automaton, and the so-called quantum computers are special cases of this. Many people like to imagine that quantum computers are kind-of-like Turing machines with an infinite number of states, but, due to the no-cloning theorem, they can't have tapes, so this is not the correct way to imagine them. The correct concept for the $$n\to\infty$$ limit of a Turing machine is of the automorphism group of a homogeneous space. The symbols/rewrite terms/grammar are just the rules that generate sequences of automorphisms that are to be applied to the space. The stopping function ("this computation has run to it's conclusion and the machine has halted and here is the answer: its 42") is an integral over a measure defined by the Borel sigma algebra over the space (you cannot use the Lesbesgue measure for this, because it is complete; see for example, abstract Wiener space for why this is a problem.) These types of systems do not fit cleanly into the conventional Chomsky hierarchy, and the computational expressive power of such systems is poorly understood. This is an active area of research, it goes under the name of descriptive set theory, where the properties of the (bold-face) Borel hierarchy are studied.

 * The above is the general sketch. I don't know of any existing wikipedia article that recaps what I wrote above. Most of what I wrote above was figured out in the 1980's, 1990's by the dynamical systems people, and is so is called "well-known" by the people who know it. But I don't know any central or seminal book or journal or "theory" that encompasses this; its sort-of scattershot and spread all about the journals. I guess maybe work on sofic shifts (shift spaces) is about as central as it gets, but that has only a poor overlap with the descriptive set theory. The quantum computing people mostly kind-of ignore all this, anyway. The reason they ignore this is ergodicity aka "noise": if you just apply a bunch of automorphisms to some homogeneous space, you will typically get a washed-out mess, aka "thermalization". The quantum computing people struggle to not thermalize. There's no way for them to move "up" the Borel hierarchy. This is kind of why L-systems are interesting: if you stick each system into a bottle, (vacuum bottle, thermos bottle) where outside forces cannot screw up the internal state, then you can perform significant and meaningful computations. This is the magic of the bilipid layer in biological cells: it isolates the inside of the cell from the external world, allowing the ribosomes and the DNA to do their thing. However, given the "origins of life" debate, and the autopoiesis hoo-haa, no one really understands how this kind of isolation actually works. Nothing in ergodic theory or thermodynamics tells you how to do that, in a formal rigorous proof-theory mathematical sense. L-systems is one of the more solid approaches that we have, right now, to that particular problem. This article is mostly a pop-sci description of L-systems; it ignores formal work. 67.198.37.16 (talk) 20:46, 11 February 2024 (UTC)

Angle Used for Koch curve
Should be 60 degrees not 90 degrees. Flaviusvulso (talk) 19:25, 24 March 2017 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 2 external links on L-system. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20020503212834/http://spanky.triumf.ca/WWW/FRACTINT/lsys/truefractal.html to http://spanky.triumf.ca/www/fractint/lsys/truefractal.html
 * Added archive https://web.archive.org/web/20031220074045/http://www.generation5.org/content/2002/lse.asp to http://www.generation5.org/content/2002/lse.asp

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 22:01, 25 May 2017 (UTC)

Hilbert curve example
Hi, the Hilbert_curve is not so didactic as here, can you add the Hilbert curve (in a more didactic steps) as more one example?

title: Talk:L-system


text:
Hedgehog

A  hedgehog  is a spiny mammal of the subfamily Erinaceinae, in the eulipotyphlan family Erinaceidae. There are seventeen species of hedgehog in five genera found throughout parts of Europe, Asia, and Africa, and in New Zealand by introduction. There are no hedgehogs native to Australia and no living species native to the Americas. However, the extinct genus Amphechinus was once present in North America.

Hedgehogs share distant ancestry with shrews (family Soricidae), with gymnures possibly being the intermediate link, and they have changed little over the last fifteen million years. Like many of the first mammals, they have adapted to a nocturnal way of life. Their spiny protection resembles that of porcupines, which are rodents, and echidnas, a type of monotreme.

Etymology
The name hedgehog came into use around the year 1450, derived from the Middle English heyghoge, from heyg, hegge ("hedge"), because it frequents hedgerows, and hoge, hogge ("hog"), from its piglike snout. Another name that is used is hedgepig.

Description
Hedgehogs are easily recognized by their spines, which are hollow hairs made stiff with keratin. Their spines are not poisonous or barbed and, unlike the quills of a porcupine, do not easily detach from their bodies. However, the immature animal's spines normally fall out as they are replaced with adult spines. This is called "quilling". Spines can also shed when the animal is diseased or under extreme stress. Hedgehogs are usually brown, with pale tips to the spines, though blonde hedgehogs are found on the Channel Island of Alderney.

Hedgehogs roll into a tight spiny ball when threatened, tucking in the furry face, feet, and belly. The hedgehog's back contains two large muscles that direct the quills. Some light-weight desert hedgehog species with fewer spines are more likely to flee or attack, ramming an intruder with the spines, rolling up only as a last resort.

Hedgehogs are primarily nocturnal, with some species also active during the day. Hedgehogs sleep for a large portion of the day under bushes, grasses, rocks, or most commonly in dens dug underground. All wild hedgehogs can hibernate, though the duration depends on temperature, species, and abundance of food.

Hedgehogs are fairly vocal, with a variety of grunts, snuffles and/or squeals.

They occasionally perform a ritual called anointing. When the animal encounters a new scent, it will lick and bite the source, then form a scented froth in its mouth and paste it on its spines with its tongue. Some experts believe this might serve to camouflage the hedgehog with the local scent, and might also lead to infection of predators poked by the spines. Anointing is sometimes also called anting after a similar behavior in birds.

Like opossums, mice, and moles, hedgehogs have some natural immunity against some snake venom through the protein erinacin in their muscles, though in such small amounts that a viper bite may still be fatal. In addition, hedgehogs are one of four known mammalian groups with natural protection against another snake venom, α-neurotoxin. Developing independently, pigs, honey badgers, mongooses, and hedgehogs all have mutations in the nicotinic acetylcholine receptor that prevent the binding of the snake venom α-neurotoxin.

The sense of smell has been little studied in the hedgehog, as the olfactory part of the mammal brain is obscured inside the neopallium. Tests have suggested that hedgehogs share the same olfactory electrical activity as cats.

Diet
Although traditionally classified in the abandoned order Insectivora, hedgehogs are omnivorous. They feed on insects, snails, frogs and toads, snakes, bird eggs, carrion, mushrooms, grass roots, berries, and melons. Afghan hedgehogs devour berries in early spring after hibernation. Hedgehogs have been observed eating cat food left outdoors for pets, but this may not be a proper food for hedgehogs in captivity.[video:1 ]

Hibernation
When a hedgehog hibernates, its normal 30-35 C body temperature decreases to 2-5 C.

Reproduction and lifespan
Hedgehog gestation lasts 35–58 days, depending on species. The average litter is 3–4 newborns for larger species and 5–6 for smaller ones. As with many animals, it is not unusual for an adult male hedgehog to kill newborn males.

Hedgehogs have a relatively long lifespan for their size. In captivity, lack of predators and controlled diet contribute to a lifespan of 8–10 years depending on size. In the wild, larger species live 4–7 years (some recorded up to 16 years), and smaller species live 2–4 years (4–7 in captivity). This compares to a mouse at 2 years and a large rat at 3–5 years.

Newborn hoglets are blind, with their quills covered by a protective membrane which dries and shrinks over several hours, and falls off after cleaning, allowing the quills to emerge.

Predators
The various species have many predators: while forest hedgehogs are prey primarily to birds (especially owls) and ferrets, smaller species like the long-eared hedgehog are prey to foxes, wolves, and mongooses. Hedgehog bones have been found in the pellets of the Eurasian eagle owl.

In Britain, the main predator is the European badger. European hedgehog populations in the United Kingdom are lower in areas with many badgers, and hedgehog rescue societies will not release hedgehogs into known badger territories. Badgers also compete with hedgehogs for food.

Domestication
The most common pet species of hedgehog are hybrids of the white-bellied hedgehog or four-toed hedgehog (Atelerix albiventris) and the smaller North African hedgehog (A. algirus, pygmy hedgehog). Other species kept as pets are the long-eared hedgehog (Hemiechinus auritus) and the Indian long-eared hedgehog (H. collaris).

it is illegal to own a hedgehog as a pet in four US states including Hawaii, Georgia, Pennsylvania, and California, as well as New York City, Washington, DC and some Canadian municipalities, and breeding licenses are required. No such restrictions exist in most European countries with the exception of Scandinavia. In Italy, it is illegal to keep wild hedgehogs as pets.

As invasive species
In areas where hedgehogs have been introduced, such as New Zealand and the islands of Scotland, the hedgehog has become a pest, lacking natural predators. In New Zealand it has decimated native species including insects, snails, lizards and ground-nesting birds, particularly shore birds.

Eradication can be troublesome. Attempts to eliminate hedgehogs from bird colonies on the Scottish islands of North Uist and Benbecula in the Outer Hebrides were met with international protest. Eradication began in 2003 with 690 hedgehogs killed, though animal welfare groups attempted rescues. By 2007, legal injunctions prohibited the killing, and in 2008, the elimination process was changed to trapping and releasing on the mainland.

In 2022, it was reported that the hedgehog population in rural Britain was declining rapidly, down by 30%-75% since 2000.

Diseases
Hedgehogs suffer many diseases common to mammals, including cancer, fatty liver disease, and cardiovascular disease.

Cancer is very common in hedgehogs. The most common is squamous cell carcinoma, which spreads quickly from bone to the organs, unlike in humans. Surgery to remove the bone tumors is impractical.

Fatty liver and heart disease are believed to be caused by bad diet and obesity. Hedgehogs will eagerly eat foods high in fat and sugar, despite a metabolism adapted for low-fat, protein-rich insects.

Hedgehogs are also highly susceptible to pneumonia, with difficulty breathing and nasal discharge, caused by the bacterium Bordetella bronchiseptica.

Hedgehogs uncommonly transmit a fungal ringworm or dermatophytosis skin infection to human handlers and other hedgehogs, caused by Trichophyton erinacei, a distinct mating group among the Arthroderma benhamiae fungi.

Hedgehogs can suffer from balloon syndrome, a rare condition in which gas is trapped under the skin from injury or infection, causing the animal to inflate. The condition is unique to hedgehogs because their skin is baggy enough to curl up. In 2017 the BBC reported a case of a male hedgehog "almost twice its natural size, literally blown up like a beach ball with incredibly taut skin". At Stapeley's Wildlife Hospital, vet Bev Panto, said, "I have seen three or four of these cases and they are very strange every time and quite shocking ... When you first see them they appear to be very big hedgehogs but when you pick them up they feel so light because they are mostly air". The British Hedgehog Preservation Society advises: "There is no single cause for this condition. The air can be removed by incising or aspirating through the skin over the back. Antibiotic cover should be given. This may be associated with lung/chest wall damage or a small external wound acting like a valve or a clostridium type infection."

Human influence
As with most small mammals living around humans, many are run over as they attempt to cross roadways. In Ireland, hedgehogs are one of the most common mammalian road fatalities. Between April 2008 and November 2010 on two stretches of road measuring 227 km and 32.5 km there were 133 recorded hedgehog fatalities. Of another 135 hedgehog carcasses collected from throughout Ireland, there were significantly more males than females collected, with peaks in male deaths occurring in May and June. Female deaths outnumbered males only in August, with further peaks in female deaths observed in June and July. It is suggested that these peaks are related to the breeding season (adults) and dispersal/exploration following independence.

Domesticated hedgehogs can get their heads stuck in tubes such as toilet paper tubes, and walk around with them. Some owners call this "tubing" and promote the behavior, providing a tube cut lengthwise to allow the hedgehog to remove it. Some hedgehogs intentionally wear tubes for hours.

Culinary and medicinal use
Hedgehogs are a food source in many cultures. They were eaten in Ancient Egypt and some recipes of the Late Middle Ages call for hedgehog meat. They are traded throughout Eurasia and Africa for traditional medicine and witchcraft. In the Middle East and especially among Bedouins, hedgehog meat is considered medicine against rheumatism and arthritis. Hedgehogs are also said to cure a variety of disorders from tuberculosis to impotence. In Morocco, inhaling the smoke of the burnt skin or bristles supposedly remedies fever, impotence, and urinary illnesses; the blood is sold as a cure for ringworm, cracked skin and warts, and the flesh is eaten as a remedy for witchcraft. Romani people still eat hedgehogs, boiled or roasted, and also use the blood and the fat as a medicine.

In 1981, British publican Philip Lewis developed a line of Hedgehog Flavoured Crisps, whose taste was apparently based on the flavourings used by Romani to bake hedgehogs. As they did not contain any actual hedgehog product, the Office of Fair Trading ordered him to change the name to Hedgehog Flavour Crisps.

Genera and species

Subfamily Erinaceinae (hedgehogs)
 * Genus Atelerix
 * Four-toed hedgehog, Atelerix albiventris
 * North African hedgehog, Atelerix algirus
 * Southern African hedgehog, Atelerix frontalis
 * Somali hedgehog, Atelerix sclateri
 * Genus Erinaceus
 * Amur hedgehog, Erinaceus amurensis
 * Southern white-breasted hedgehog, Erinaceus concolor
 * European hedgehog, Erinaceus europaeus
 * Northern white-breasted hedgehog, Erinaceus roumanicus
 * Genus Hemiechinus
 * Long-eared hedgehog, Hemiechinus auritus
 * Indian long-eared hedgehog, Hemiechinus collaris
 * Genus Mesechinus
 * Daurian hedgehog, Mesechinus dauuricus
 * Hugh's hedgehog, Mesechinus hughi
 * Small-toothed forest hedgehog, Mesechinus miodon
 * Gaoligong forest hedgehog, Mesechinus wangi
 * Genus Paraechinus
 * Desert hedgehog, Paraechinus aethiopicus
 * Brandt's hedgehog, Paraechinus hypomelas
 * Indian hedgehog, Paraechinus micropus
 * Bare-bellied hedgehog, Paraechinus nudiventris

Society and culture
In worldwide folklore, hedgehogs are associated with intelligence and wisdom (Asia, Europe), and magic (Africa).

title: Hedgehog