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Condition number

In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given $$f(x) = y,$$ one is solving for x, and thus the condition number of the (local) inverse must be used.

The condition number is derived from the theory of propagation of uncertainty, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of the matrix. More generally, condition numbers can be defined for non-linear functions in several variables.

A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. In non-mathematical terms, an ill-conditioned problem is one where, for a small change in the inputs (the independent variables) there is a large change in the answer or dependent variable. This means that the correct solution/answer to the equation becomes hard to find. The condition number is a property of the problem. Paired with the problem are any number of algorithms that can be used to solve the problem, that is, to calculate the solution. Some algorithms have a property called backward stability; in general, a backward stable algorithm can be expected to accurately solve well-conditioned problems. Numerical analysis textbooks give formulas for the condition numbers of problems and identify known backward stable algorithms.

As a rule of thumb, if the condition number $$\kappa(A) = 10^k$$, then you may lose up to $$k$$ digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods. However, the condition number does not give the exact value of the maximum inaccuracy that may occur in the algorithm. It generally just bounds it with an estimate (whose computed value depends on the choice of the norm to measure the inaccuracy).

General definition in the context of error analysis
Given a problem $$f$$ and an algorithm $$\tilde{f}$$ with an input $$x$$ and output $$\tilde{f}(x),$$ the error is $$\delta f(x) := f(x) - \tilde{f}(x),$$ the absolute error is $$\|\delta f(x)\| = \left\|f(x) - \tilde{f}(x)\right\|$$ and the relative error is $$\|\delta f(x)\| / \|f(x)\| = \left\|f(x) - \tilde{f}(x)\right\| / \|f(x)\|.$$

In this context, the absolute condition number of a problem $$f$$ is

 * $$\lim_{\varepsilon \rightarrow 0^+}\, \sup_{\|\delta x\| \,\leq\, \varepsilon} \frac{\|\delta f(x)\|}{\|\delta x\|}$$

and the relative condition number is

 * $$\lim_{\varepsilon \rightarrow 0^+}\, \sup_{\|\delta x \| \,\leq\, \varepsilon} \frac{\|\delta f(x)\| / \|f(x)\|}{\|\delta x\| / \|x\|}.$$

Matrices
For example, the condition number associated with the linear equation Ax = b gives a bound on how inaccurate the solution x will be after approximation. Note that this is before the effects of round-off error are taken into account; conditioning is a property of the matrix, not the algorithm or floating-point accuracy of the computer used to solve the corresponding system. In particular, one should think of the condition number as being (very roughly) the rate at which the solution x will change with respect to a change in b. Thus, if the condition number is large, even a small error in b may cause a large error in x. On the other hand, if the condition number is small, then the error in x will not be much bigger than the error in b.

The condition number is defined more precisely to be the maximum ratio of the relative error in x to the relative error in b.

Let e be the error in b. Assuming that A is a nonsingular matrix, the error in the solution A−1b is A−1e. The ratio of the relative error in the solution to the relative error in b is

 * $${\frac{\left\|A^{-1} e\right\|}{\left\|A^{-1} b\right\|}}/{\frac{\|e\|}{\|b\|}} = \frac{\left\|A^{-1} e\right\|}{\|e\|} \frac{\|b\|}{\left\|A^{-1} b\right\|}.$$

The maximum value (for nonzero b and e) is then seen to be the product of the two operator norms as follows:

 * $$\begin{align}

\max_{e,b \neq 0} \left\{ \frac{\left\| A^{-1}e \right\|}{\| e \|} \frac{\| b \|}{\left\| A^{-1}b \right\|} \right\} &= \max_{e \neq 0} \left\{\frac{\left\| A^{-1}e\right\| }{\| e\|} \right\} \, \max_{b \neq 0} \left\{ \frac {\| b \|}{\left\| A^{-1}b \right\|} \right\} \\ &= \max_{e \neq 0} \left\{\frac{\left\| A^{-1}e\right\|}{\| e \|}\right\} \, \max_{x \neq 0} \left \{\frac {\| Ax \| }{\| x \|} \right\} \\ &= \left\| A^{-1} \right \| \, \|A\|. \end{align}$$

The same definition is used for any consistent norm, i.e. one that satisfies

 * $$\kappa(A) = \left\| A^{-1} \right\| \, \left\| A \right\| \ge \left\| A^{-1} A \right\| = 1.$$

When the condition number is exactly one (which can only happen if A is a scalar multiple of a linear isometry), then a solution algorithm can find (in principle, meaning if the algorithm introduces no errors of its own) an approximation of the solution whose precision is no worse than that of the data.

However, it does not mean that the algorithm will converge rapidly to this solution, just that it will not diverge arbitrarily because of inaccuracy on the source data (backward error), provided that the forward error introduced by the algorithm does not diverge as well because of accumulating intermediate rounding errors.

The condition number may also be infinite, but this implies that the problem is ill-posed (does not possess a unique, well-defined solution for each choice of data; that is, the matrix is not invertible), and no algorithm can be expected to reliably find a solution.

The definition of the condition number depends on the choice of norm, as can be illustrated by two examples.

If $$\|\cdot\|$$ is the matrix norm induced by the (vector) Euclidean norm (sometimes known as the L2 norm and typically denoted as $$\|\cdot\|_2$$), then

 * $$\kappa(A) = \frac{\sigma_\text{max}(A)}{\sigma_\text{min}(A)},$$

where $$\sigma_\text{max}(A)$$ and $$\sigma_\text{min}(A)$$ are maximal and minimal singular values of $$A$$ respectively. Hence:

 * If $$A$$ is normal, then $$\kappa(A) = \frac{\left|\lambda_\text{max}(A)\right|}{\left|\lambda_\text{min}(A)\right|},$$ where $$\lambda_\text{max}(A)$$ and $$\lambda_\text{min}(A) $$ are maximal and minimal (by moduli) eigenvalues of $$A$$ respectively.
 * If $$A$$ is unitary, then $$\kappa(A) = 1.$$

The condition number with respect to L2 arises so often in numerical linear algebra that it is given a name, the condition number of a matrix.

If $$\|\cdot\|$$ is the matrix norm induced by the $L^\infty$ (vector) norm and $$A$$ is lower triangular non-singular (i.e. $$a_{ii} \ne 0$$ for all $$i$$), then

 * $$\kappa(A) \geq \frac{\max_i\big(|a_{ii}|\big)}{\min_i\big(|a_{ii}|\big)}$$

recalling that the eigenvalues of any triangular matrix are simply the diagonal entries.

The condition number computed with this norm is generally larger than the condition number computed relative to the Euclidean norm, but it can be evaluated more easily (and this is often the only practicably computable condition number, when the problem to solve involves a non-linear algebra, for example when approximating irrational and transcendental functions or numbers with numerical methods).

If the condition number is not significantly larger than one, the matrix is well-conditioned, which means that its inverse can be computed with good accuracy. If the condition number is very large, then the matrix is said to be ill-conditioned. Practically, such a matrix is almost singular, and the computation of its inverse, or solution of a linear system of equations is prone to large numerical errors.

A matrix that is not invertible is often said to have a condition number equal to infinity. Alternatively, it can be defined as $$\kappa(A) = \|A\| \|A^\dagger\|$$, where $$A^\dagger$$ is the Moore-Penrose pseudoinverse. For square matrices, this unfortunately makes the condition number discontinuous, but it is a useful definition for rectangular matrices, which are never invertible but are still used to define systems of equations.

Nonlinear
Condition numbers can also be defined for nonlinear functions, and can be computed using calculus. The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.

One variable
The condition number of a differentiable function $$f$$ in one variable as a function is $$\left|xf'/f\right|$$. Evaluated at a point $$x$$, this is

 * $$\left|\frac{xf'(x)}{f(x)}\right|=\left|\frac{(\log f)'}{(\log x)'}\right|.$$

Note that this is the absolute value of the elasticity of a function in economics.

Most elegantly, this can be understood as (the absolute value of) the ratio of the logarithmic derivative of $$f$$, which is $$(\log f)' = f'/f$$, and the logarithmic derivative of $$x$$, which is $$(\log x)' = x'/x = 1/x$$, yielding a ratio of $$xf'/f$$. This is because the logarithmic derivative is the infinitesimal rate of relative change in a function: it is the derivative $$f'$$ scaled by the value of $$f$$. Note that if a function has a zero at a point, its condition number at the point is infinite, as infinitesimal changes in the input can change the output from zero to positive or negative, yielding a ratio with zero in the denominator, hence infinite relative change.

More directly, given a small change $$\Delta x$$ in $$x$$, the relative change in $$x$$ is $$[(x + \Delta x) - x] / x = (\Delta x) / x$$, while the relative change in $$f(x)$$ is $$[f(x + \Delta x) - f(x)] / f(x)$$. Taking the ratio yields

 * $$\frac{[f(x + \Delta x) - f(x)] / f(x)}{(\Delta x) / x} = \frac{x}{f(x)} \frac{f(x + \Delta x) - f(x)}{(x + \Delta x) - x} = \frac{x}{f(x)} \frac{f(x + \Delta x) - f(x)}{\Delta x}.$$

The last term is the difference quotient (the slope of the secant line), and taking the limit yields the derivative.

Condition numbers of common elementary functions are particularly important in computing significant figures and can be computed immediately from the derivative. A few important ones are given below:

Several variables
Condition numbers can be defined for any function $$f$$ mapping its data from some domain (e.g. an $$m$$-tuple of real numbers $$x$$) into some codomain (e.g. an $$n$$-tuple of real numbers $$f(x)$$), where both the domain and codomain are Banach spaces. They express how sensitive that function is to small changes (or small errors) in its arguments. This is crucial in assessing the sensitivity and potential accuracy difficulties of numerous computational problems, for example, polynomial root finding or computing eigenvalues.

The condition number of $$f$$ at a point $$x$$ (specifically, its relative condition number ) is then defined to be the maximum ratio of the fractional change in $$f(x)$$ to any fractional change in $$x$$, in the limit where the change $$\delta x$$ in $$x$$ becomes infinitesimally small:

 * $$\lim_{\varepsilon \to 0^+} \sup_{\|\delta x\| \leq \varepsilon} \left[ \left. \frac{\left\|f(x + \delta x) - f(x)\right\|}{\|f(x)\|} \right/ \frac{\|\delta x\|}{\|x\|} \right],$$

where $$\|\cdot\|$$ is a norm on the domain/codomain of $$f$$.

If $$f$$ is differentiable, this is equivalent to:

 * $$\frac{\|J(x)\|}{\|f(x) \| / \|x\|},$$

where $J(x)$ denotes the Jacobian matrix of partial derivatives of $$f$$ at $$x$$, and $$\|J(x)\|$$ is the induced norm on the matrix.

title: Condition number


text:
Talk:Cheddar, Somerset

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: Moved the two requested pages. I did not perceive a clear verdict on whether Cheddar cheese should be moved to Cheddar (i.e. that the cheese is the primary topic). That could be settled by another move proposal if people still think it's a good idea after reading over all the comments here. EdJohnston (talk) 22:12, 4 March 2014 (UTC)

– I can't believe this has never been brought up in the (checking here) almost thirteen years of this article's existence. (Seriously, look at the history. It was created just six days after 9/11.) The cheese is almost certainly what most people looking for when they search "Cheddar"; pretty much everywhere outside the UK (and arguably many if not most places within the UK as well, especially if the people in question don't live in the southwest part of the country), whenever the word Cheddar is used, it almost always refers to the cheese. In fact, I don't think it would be too much of a stretch to argue that the disambig page should be kept as is and the Cheddar article should become a redirect to the cheese. Literally millions of people worldwide eat things with Cheddar cheese, while not nearly as many people would know about a relatively small town in England (especially with several larger and better-known cities and towns nearby, such as Bath) without having looked it up first. 128.206.196.188 (talk) 18:49, 24 February 2014 (UTC)
 * Cheddar → Cheddar, Somerset
 * Cheddar (disambiguation) → Cheddar
 * Support If there's a WP:PRIMARYTOPIC, it's probably Cheddar cheese. It's certainly not this village. Last month, the cheese received nearly five times as many views as the village, and some of the village's 5000 or so visitors were surely looking for the cheese or another topic. While this title has been stable, that doesn't necessarily mean it's where it should be. Well done on bringing this up. For now, the move as proposed is fine, but if there's consensus here to make the cheese primary topic, I wouldn't oppose that. --BDD (talk) 19:28, 24 February 2014 (UTC)
 * Comment It is extremely difficult to provide any evidence about what people were looking for when they visit any article. They may have been looking for the better known Cheddar Gorge which, if we are using page views as an indicator, had 13263 views in the last 90 days (as opposed to 15378 for the village and 71215 for the cheese). Ghits and google searches are not much better as an indicator but if I search for "Cheddar" on Google the first is the village web site followed by the gorge, then the wp article on the village before the wp article on the cheese.— Rod talk 20:20, 24 February 2014 (UTC)
 * Yes, I'm afraid Google is becoming less and less a useful tool for these sorts of discussions as they try to personalize results. I'm in the US, and my top results for are our page on the cheese, an American restaurant chain, and a smartphone app. In fact, I went through the first five pages of results without seeing a mention of the village—not even our article, which is surprising. I get the same results in a private browsing window, which makes me think this is IP based.
 * Again, I'm not necessarily arguing that the cheese is primary right now, though I suspect it could be. Incidentally, when I performed the above Google search I ran across this recent Boston Globe piece saying that Cheddar cheese is "the most popular variety of cheese worldwide, and also the most studied." --BDD (talk) 21:06, 24 February 2014 (UTC)

 * Oppose The same argument could probably be made for just about every cheese named after its town or county of origin, along with wines and probably a host of other foodstuffs. If people don't know the difference between Cheddar and Cheddar Cheese, or even that it might just be named after a place or person, this is a great learning opportunity for them. Wikipedia provides plenty of guidance to the right article; type Chedd into the search box and Cheddar cheese is right there in the drop down menu for you to choose from. If WP:PRIMARYTOPIC really suggests that a disruptive and illogical move like this is a good idea, then perhaps it needs rethinking. --Derek Andrews (talk) 21:09, 24 February 2014 (UTC)
 * Not necessarily. It's going to depend on the usual metrics, such as page views. When I hear "Kalamata" I think olives, but we have Kalamata, on the city, as primary versus Kalamata (olive) (interestingly, the olive does get more page views, but it's close—nowhere near the large disparity here). Champagne is about the drink, though that's a slightly different case. --BDD (talk) 21:14, 24 February 2014 (UTC)
 * BDD, if someone was looking for Kalamata olives how could they possibly arrive at the wrong article? In ictu oculi (talk) 21:56, 24 February 2014 (UTC)
 * I'm not sure I follow. A reader could search "Kalamata" knowing about Kalamata olives but not the city. --BDD (talk) 22:02, 24 February 2014 (UTC)
 * I mean how would they search Kalamata knowing about Kalamata olives but not the city and not end up at the olive article? In ictu oculi (talk) 22:07, 24 February 2014 (UTC)
 * Are we talking about on the first try or eventually? Sure, our hypothetical reader could end up on the city article and have to follow the hatnote to the olive. But if the olive were primary, this wouldn't be a satisfactory arrangement. Hatnotes are navigational aids, but they're also inconveniences, however minor. If we can easily direct more readers straight to the article they're looking for without having to go through hatnotes, we should. --BDD (talk) 22:15, 24 February 2014 (UTC)
 * I mean first try, why would they choose "Kalamata" over "Kalamata olive" if they are looking for Kalamata olive? In ictu oculi (talk) 22:53, 24 February 2014 (UTC)
 * If they didn't know that Kalamata referred to anything but an olive. It may sound silly to us, not all readers will know this. More germane to this discussion, there will be readers who know Cheddar is a cheese but not a village. And while I laud the intent of Derek Andrews's "learning opportunity" on the matter, I don't think complicating navigation is the right way to do that. The first paragraph of Cheddar cheese mentions the cheese's origins here anyway. --BDD (talk) 23:41, 24 February 2014 (UTC)
 * User:BDD that's exactly my point, say your reader who doesn't know Kalamata is a place enters Kalamata in either Google or the top right hand search box, they will see:
 * Kalamata
 * Kalamata F.C.
 * Kalamata International Airport
 * Kalamata olive
 * .......so why if they think Kalamata is an olive would they then choose "Kalamata" over "Kalamata (olive)". It isn't a challenging learning opportunity or an intelligence test, it's the difference between a human being and being a piece of wood. Why would a reader avoid "Kalamata olive" and select "Kalamata" if he/she is searching for Kalamata olive? In ictu oculi (talk) 18:17, 25 February 2014 (UTC)
 * If there's one thing I've learned from all the time I spend at RfD, it's that readers can get to articles in many ways. They might type "Kalamata" directly into a URL, or into a plug-in for their web browser that doesn't provide search suggestions (such as the one I use for Chrome). The suggestions in Wikipedia's own search box are certainly helpful, but we do a disservice to our readers by assuming that's the only way they find pages. --BDD (talk) 18:19, 25 February 2014 (UTC)
 * I don't understand that, in Chrome the url space has dual functions as the Explorer search box. If I enter [Wikipedia Kalamata] in the URL bar and press ENTER I get exactly the same result as in Explorer or Firefox, just direct to Google. I still have the choice:
 * Kalamata
 * Kalamata F.C.
 * Kalamata International Airport
 * Kalamata olive
 * How can a Chrome User disable this so he/she only sees one choice? In ictu oculi (talk) 19:07, 25 February 2014 (UTC)
 * I'm afraid we're really straying off topic now, but I have a plug-in (extension, technically) in Chrome that appears as a magnifying glass icon to the right of the URL bar. Clicking it brings up several sites to search, such as Google, Amazon, and Wikipedia. I don't know its name offhand and am not using Chrome right now, but I will most often type my search term in there to bring up something on Wikipedia. I'm not sure how to explain this better except to note that it's not using the URL bar. Firefox has a similar search function built in (again, to the right of the URL bar), although this one taps into suggestions similar to the native Wikipedia search bar. --BDD (talk) 19:14, 25 February 2014 (UTC)
 * Well it's not really off topic, since if some Users have installed plug ins that prevent them seeing the choices Google/Chrome/Explorer/Firefox and WP's own search box provide it's worth knowing about it. But personally I think our titling should work on the basis of readers who haven't disabled their browsers ability to see more than one title. It does explain though your own previous comments in other RMs, so thanks for explaining. In ictu oculi (talk) 20:52, 25 February 2014 (UTC)

 * Oppose - The Cheddar Gorge, the Cheddar Valley, the Cheddar Valley line and yes Cheddar cheese all take their name from Cheddar - though the cheese is apparently named after farms all along the Cheddar Valley not the village.
 * As regards readers who aren't aware that Category:English cheeses are all named after places, yes they will arrive at "X" geographical places not "X cheese" if they deliberately select "X" while ignoring "X cheese" showing in the right hand search box, or if they click "X" while ignoring "X cheese" in Google results. In which case they will benefit from learning basic reading skills that "cheese" indicates an article on a cheese. The fact that the place Cheddar (and the Cheddar Valley) has much lower hits than Cheddar cheese shows that naming the place after the place and the cheese after the cheese is working. Consider the alternative: if Cheddar the place was getting gazillions of hits then we'd know that calling the cheese a "cheese" wasn't working and then we would change it, but it is working, so why hide the cheese?
 * If we move these we need to hide the word cheese on all the articles in the category to prevent readers seeing the word cheese on all the cheese. The reason a move has never been brought up in 13 years is because calling a cheese a "cheese" works. In ictu oculi (talk) 21:49, 24 February 2014 (UTC)
 * This request isn't about renaming Cheddar cheese, nor would I support such a proposal. --BDD (talk) 22:02, 24 February 2014 (UTC)
 * No, but a dab is just as pointless. The only way someone looking for Cheddar cheese could arrive at Cheddar (disambiguation) is if they couldn't recognize the word "cheese" in Cheddar cheese in the search results. In ictu oculi (talk) 22:07, 24 February 2014 (UTC)
 * Ok, that's fair. Despite my insistence that I'm not arguing for the cheese is primary, I think I've been doing just that, assuming that this move would have "Cheddar" redirecting to the cheese. That's not what's proposed, though. --BDD (talk) 22:15, 24 February 2014 (UTC)
 * So, lets get this right. Cheddar is going to become a disambiguation page? What exactly is this going to achieve? Currently if they end up at the article about the village instead of the cheese, they see a link to Cheddar Cheese, effectively the same result. --Derek Andrews (talk) 00:09, 25 February 2014 (UTC)

 * Support. I'd been waiting for pageview statistics to accrue before proposing this move, but here we go, I guess. At the top of the article, the hatnote links to "Cheddar (cheese)" instead of Cheddar cheese"; basically no other article links to exactly that 'title'. This article on the city has been viewed 4187 times in February, and well over a tenth of the readers have immediately clicked the link to cheddar (cheese) . Considering most people arrive here by direct links and many of them probably arrived from cheddar cheese in the first place (by FAR the most important topic named "cheddar" with over 11000 views so far this month), that's a pretty good signal to me that a lot of people are looking for the cheese at "cheddar". I sure did once upon a time. This village is not a very important one (see the move request at [[Plymouth] I opposed; Plymouth's not going anywhere because it's a major British city. Cheddar is a town where a type of cheese was developed) as far as settlements go. I would also like to note that "X was named after Y" is not and has never been a naming criteria at Wikipedia. As per clear WP:PRIMARYTOPIC, I support moving Cheddar to Cheddar, Somerset. I oppose moving the disambiguation page, however; cheddar cheese is the primary topic and all other uses can be dealt with via a hatnote (with, of course, the village prominently and directly linked to in the hatnote). Red Slash 01:13, 25 February 2014 (UTC)
 * I don't follow your argument about the hatnote. Surely this is showing that firstly the hatnote is wrong and should be pointing directly to Cheddar cheese, and secondly that Cheddar cheese is a thing defined by two words and not one. --Derek Andrews (talk) 01:54, 25 February 2014 (UTC)
 * What next? Brie cheese?  "Cheddar cheese" is just as redundant as "brie cheese".  --B2C 17:26, 25 February 2014 (UTC)
 * This is a comment that can be empirically tested, is "cheese" redundant?
 * added for clarity Cheddar +meal or food or recipe or eat = 454,000 results total pool
 * Cheddar +meal or food or recipe or eat -cheese 36,800 results
 * Cheddar +meal or food or recipe or eat +cheese 323,000 results
 * Only 11.39% of the sample of books using Cheddar with any 1 of "meal"/"food"/"recipe"/"eat" considered "cheese" redundant. While 88.61% of the same base sample of books decided to also use the word "cheese". If "cheese" is redundant, why do 88.61% of the sample say both "Cheddar" and "cheese". of those 253,000 use the two words together "Cheddar cheese" - which is 78.32%. 78.32% is pretty high for something which is "redundant". In ictu oculi (talk) 18:37, 25 February 2014 (UTC)
 * I think your search string is wrong. Try this result. Basically, you should only exclude those that mention "cheddar cheese" as a phrase instead of "cheese". With the modified search string, we get 152,000 results related to cheddar cheese but doesn't mention "cheddar cheese" or 47%. Definitely not insignificant. —seav (talk) 19:52, 25 February 2014 (UTC)
 * I've spelled out the full formula and added in the pool 454,000 (which doesn't total of course). Your search on "cheddar cheese" is the inverse of the search I did. But still produces the same result. 253,000 use the two words together "Cheddar cheese" - 152 don't. The majority of sources use "cheddar cheese" and the large majority still mention cheese. The purpose of the test was to show that Born2cycle's statement that "cheese" is "redundant" is not true, which you have by posting this search confirmed. In ictu oculi (talk) 20:47, 25 February 2014 (UTC)
 * First, I did not make the "statement that 'cheese' is 'redundant'", so any effort to prove that statement false is a waste-of-time strawman argument. Secondly, what I did say is that "cheddar cheese" is just as redundant as "brie cheese".
 * cheddar meal OR food OR recipe OR eat +"cheddar cheese" About 252,000 results (0.19 seconds)
 * cheddar meal OR food OR recipe OR eat +"cheddar" -"cheddar cheese" About 152,000 results (0.74 seconds)
 * brie meal OR food OR recipe OR eat +"brie cheese" About 5,460 results (0.45 seconds)
 * brie meal OR food OR recipe OR eat +"brie" -"brie cheese" About 74,300 results (0.72 seconds)
 * Okay, so I exaggerated. "Cheddar cheese" is not just as redundant as "brie cheese" (brie alone is used 13 times as often as "brie cheese), but enough use just "cheddar" (alone) to demonstrate many recognize "cheddar cheese" to be redundant.  People use "cheddar" (alone) about 60% of the time that "cheddar cheese" is used.  --B2C 23:03, 25 February 2014 (UTC)

 * Support. I don't think it's going out on a limb to say that the cheese (or even Cheddar Gorge) is what more people would expect to find when looking for "cheddar." Thus, there is no primary topic and disambiguation is needed.  Hot Stop   01:59, 25 February 2014 (UTC)
 * Addendum: I would like to add that I would fully support either redirecting cheddar to cheddar cheese, or move the cheese's article to the undisambiguated cheddar article.  Hot Stop   15:38, 27 February 2014 (UTC)

 * Support per WP:PRIMARYTOPIC. I would move Cheddar cheese to Cheddar, but the dab at Cheddar is an improvement over the current situation.  --B2C 05:55, 25 February 2014 (UTC)
 * Support clear UKPLACEBIAS. The cheese is clearly the primary topic. Just as Champagne is a region in France, but the sparkling wine is the primary topic. It doesn't matter what came first, otherwise we'll be arguing Boston, Lincolnshire should replace Boston, Massachusetts -- 70.50.151.11 (talk) 07:35, 25 February 2014 (UTC)
 * Support. Hchc2009 (talk) 08:12, 25 February 2014 (UTC)
 * Support. I looked at Google Books results and majority of the results were about the cheese. I personally think that the cheese deserves the WP:PRIMARYTOPIC status, but the proposed DAB page is better than the status quo. —seav (talk) 16:12, 25 February 2014 (UTC)
 * User:Seav, how exactly? See User:Derek Andrews' question above. How is 1 extra click via a dab list better than 1 extra click through a hat note. All this RM is proposing is a reroute of the extra click to take a click detour (click bypass) avoiding the place that makes the cheese. Why is a detour via the dab better than a detour via the place? In ictu oculi (talk) 18:37, 25 February 2014 (UTC)
 * It's better because it follows the WP:AT policy and the WP:DAB guidelines better. I think your point regarding detours is a red herring because you are assuming that users get to arrive at articles in a particular manner. I don't see anything in Wikipedia's guidelines and policies that says routing via hatnotes is better than DAB pages other than one recommendation mentioned in WP:TWODABS, which does not apply in this case. —seav (talk) 19:00, 25 February 2014 (UTC)
 * But that that's the problem, it doesn't follow WP:AT policy better. WP:CRITERIA has 5 parts: Recognizability, Naturalness, Conciseness, Consistency. This move fails on 4 out of 5 reducing 1 Recognizability, 2 Naturalness, 3 Precision, 5 Consistency at the expense of only Criteria No. 4 Conciseness.
 * We've already tested these criteria for the point of focus - what is the cheese known as - and got a of those 253,000 use the two words together "Cheddar cheese" - which is 78.32% support for Naturalness for the current status of "Cheddar cheese".
 * That only leaves application of WP:AT to the place, and yes there is some increase in 3 Precision for Cheddar → Cheddar, Somerset, and arguably 1 Recognizability for the small number of people looking for the place but need to see "Somerset" added. (wouldn't have helped me as I thought Cheddar was Wiltshire) In ictu oculi (talk) 19:17, 25 February 2014 (UTC)
 * The proposed move are 2 things: First is "Cheddar" → "Cheddar, Somerset". This first move follows WP:DAB (specifically, WP:PRIMARYTOPIC) better since the evidence shared above points towards the village not deserving the primary topic status among all things named "Cheddar".
 * Second is "Cheddar (disambiguation)" → "Cheddar". If there is consensus that the village title move is warranted, then this second move is the most obvious result. (There may be the case that the cheese deserves the primary topic status and so the second move could instead be "Cheddar cheese" → "Cheddar" [or alternately, "Cheddar" redirects to "Cheddar cheese"], but it's still debatable that the cheese is indeed deserving of the primary topic status.) —seav (talk) 19:37, 25 February 2014 (UTC)

 * Oppose I need to inject us with a large dose of some common sense! - if someone were to look up cheddar cheese, they would blatantly search for Cheddar cheese and not the town of Cheddar. For historical references, cheddar cheese takes its name from this town and even to this day they are left to mature in the Cheddar Gorge. Renaming Cheddar cheese just to Cheddar wouldn't sound right - the town is considerably older and more historic than the cheese and the cheese takes its name from this town! The same thing can be said about Sandwich, Kent where again the town is more historic than the actual sandwich (the name of the town gave its name to the sandwich). This article is much broader than the cheese itself and has its own gorge, railway line and valley. Renaming this article in respect for the cheese just wouldn't make any sense. ☠ Jag  uar  ☠ 21:46, 25 February 2014 (UTC)
 * But Sandwich is about the food, not the town. Are you saying we should rename those articles? --BDD (talk) 22:56, 25 February 2014 (UTC)
 * If I was searching for the cheese, I would search with cheddar, not cheddar cheese. --B2C 23:06, 25 February 2014 (UTC)
 * Why do you say, "Renaming this article in respect for the cheese..."? Respect?  Respect has not been mentioned by anyone supporting this proposal.  Respect has nothing to do with it, nor should it.  As far as I can tell, "respect" is not one of the criteria we use to decide how to title articles. Is "respect" for the city why you oppose the move?  --B2C 04:16, 26 February 2014 (UTC)
 * I'm sorry, I didn't mean to say it like that. I think I said 'respect' as a figure of speech. I also would like to add that I don't see the point even if this article does get moved as Cheddar would only get linked to a disambiguation page - it defeats the object of the whole discussion as Cheddar cheese would not be moved or referred to just Cheddar. If we move this page to a disambiguation then Sandwich should also logically be moved to a disambiguation. ☠ Jag  uar  ☠ 13:40, 26 February 2014 (UTC)
 * , the issue on the table right now is whether this use of Cheddar is primary or not. If it is, then it should remain. If it's not, then the title needs to be disambiguated, and the dab page moved here, at least for the time being. If the result of this discussion is that this use is not primary, and the dab page is moved to Cheddar, then we can address the question of whether the cheese is primary or not.  The other way to have done it is to propose moving the cheese rather than the dab page to the base name, but then both questions would have to be answered affirmatively for the move to go through.  There is no easy way to post both questions in an RM in a way that the right moves are made accordingly.  So doing it one question/step at a time makes sense.  --B2C 19:17, 26 February 2014 (UTC)

 * Support Although I too would prefer the cheese at the base name as primary topic. benmoore 20:28, 26 February 2014 (UTC)
 * Support', though like Ben, I would prefer the undisambiguated name should point to the cheese, with a hatnote going to the disambiguation page. But clearly and regardless, from a worldwide perspective the town is not the primary topic (which explicitly rejects the argue kent that the town had the name first), and the current setup is a case of anglocentrism. oknazevad (talk) 14:53, 27 February 2014 (UTC)
 * I oppose making "Cheddar" a DAB page, but I support the move if the cheese is moved to "Cheddar". Srnec (talk) 03:59, 1 March 2014 (UTC)
 * Strong oppose making cheese the primary topic, weak oppose primary dab. The evidence above and at google books is clear that the collocation of "Cheddar" and "Cheese" is by far the most common way of referring to the cheese when cheese has not already been established as the context of the discussion. Whether the village or the dab should be primary is less clear, but many, perhaps most, people looking for the village will look for it at "Cheddar". It is not at all clear to me that any of the other options are commonly referred to as "Cheddar" alone - certainly only the village and the dab page could reasonably be located at the undisambiguated title and given this I think the village primary. Given that nobody has raised this in thirteen years is a big clue to there not being a significant problem with the current setup. Thryduulf (talk) 14:37, 3 March 2014 (UTC)
 * Support. Clearly if there were to be a primary topic it would likely be the cheese, but I'm not sure there is one.  So the proposal works well.  Vegaswikian (talk) 22:45, 3 March 2014 (UTC)
 * Support.  "Cheddar" is very easily misrecognized as the cheese, and the town is better recognized as Cheddar, Somerset. --SmokeyJoe (talk) 09:33, 4 March 2014 (UTC)
 * Support moves as nominated. The town is one of the less significant uses of the name. The two most prominent uses are cheddar cheese and the Cheddar Gorge, and neither is primary over all the other uses. -- Brown HairedGirl (talk) • (contribs) 09:46, 4 March 2014 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

GA check
I am working my way through the Good articles listed at Places; having a quick look to see if they still meet the Good article criteria. I have landed on this article. After I've had a quick look, I'll leave a note here indicating if I have concerns or not.

In general, I see the process as this: 1) Give the article a quick look to see if there are obvious issues: maintenance tags, unsourced sections, excessive media, etc, resolving any minor issues as I do so; 2) If I have concerns, open a GAR to see how serious those concerns are, resolving them myself if they are not serious; 3) If during the GAR I feel that there is significant work to be done (more than I can or am willing to do myself), I will put the GAR on hold and notify the main contributors.

My aim and intention is to keep the article listed - I would rather the article was improved and kept listed than the article is delisted. Where a delisting seems likely due to the amount or nature of work needed being greater than I am able to do alone, and the main contributors are unavailable or unable for whatever reason to do the work, then appropriate WikiProjects will be notified at least seven days before a delisting would take place.  SilkTork  ✔Tea time  21:21, 3 May 2014 (UTC)

 * ✅ OK. No issues.  SilkTork  ✔Tea time  21:33, 3 May 2014 (UTC)

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RM for Cheddar
Related to this article and the discussion that took place here in 2014, there is a requested move at Talk:Cheddar, proposing to move Cheddar to Cheddar (disambiguation) and make Cheddar redirect to Cheddar cheese. Please participate there if interested. Adumbrativus (talk) 08:31, 27 July 2022 (UTC)

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

title: Talk:Cheddar, Somerset


text:
Cheddar cheese

Cheddar cheese (or simply cheddar) is a natural cheese that is relatively hard, off-white (or orange if colourings such as annatto are added), and sometimes sharp-tasting. It originates from the English village of Cheddar in Somerset, southwest England.

Cheddar is produced all over the world, and cheddar cheese has no Protected Designation of Origin (PDO). In 2007, the name West Country Farmhouse Cheddar was registered in the European Union and (after Brexit) the United Kingdom, defined as cheddar produced from local milk within Somerset, Dorset, Devon and Cornwall and manufactured using traditional methods. Protected Geographical Indication (PGI) was registered for Orkney Scottish Island Cheddar in 2013 in the EU, which also applies under UK law.

Globally, the style and quality of cheeses labelled as cheddar varies greatly, with some processed cheeses packaged as "cheddar". Cheeses similar to Red Leicester are sometimes marketed as "red cheddar".

Cheddar is the most popular cheese in the UK, accounting for 51% of the country's £1.9 billion annual cheese market. It is the second-most popular cheese in the United States behind mozzarella, with an average annual consumption of 10 lb per capita. The United States produced approximately 3000000000 lb of cheddar in 2014, and the UK produced 258000 LT in 2008.

History
Cheddar cheese originates from the village of Cheddar in Somerset, southwest England. Cheddar Gorge on the edge of the village contains a number of caves, which provided the ideal humidity and steady temperature for maturing the cheese. Cheddar traditionally had to be made within 30 mi of Wells Cathedral.

The 19th-century Somerset dairyman Joseph Harding was central to the modernisation and standardisation of cheddar. For his technical innovations, promotion of dairy hygiene, and volunteer dissemination of modern cheese-making techniques, Harding has been dubbed "the father of cheddar". Harding introduced new equipment to the process of cheese-making, including his "revolving breaker" for curd cutting; the revolving breaker saved much manual effort in the cheese-making process. The "Joseph Harding method" was the first modern system for cheddar production based upon scientific principles. Harding stated that cheddar cheese is "not made in the field, nor in the byre, nor even in the cow, it is made in the dairy". Together, Joseph Harding and his wife introduced cheddar in Scotland and North America, while his sons Henry and William Harding were responsible for introducing cheddar cheese production to Australia and facilitating the establishment of the cheese industry in New Zealand, respectively.

During the Second World War and for nearly a decade thereafter, most of the milk in Britain was used to make a single kind of cheese nicknamed "government cheddar" as part of the war economy and rationing. As a result, almost all other cheese production in the country was wiped out. Before the First World War, more than 3,500 cheese producers were in Britain; fewer than 100 remained after the Second World War.

According to a United States Department of Agriculture researcher, cheddar is the world's most popular cheese and is the most studied type of cheese in scientific publications.

Process
During the manufacture of cheddar, the curds and whey are separated using rennet, an enzyme complex normally produced from the stomachs of newborn calves, while in vegetarian or kosher cheeses, bacterial, yeast or mould-derived chymosin is used.

"Cheddaring" refers to an additional step in the production of cheddar cheese where, after heating, the curd is kneaded with salt, cut into cubes to drain the whey, and then stacked and turned. Strong, extra-mature cheddar, sometimes called vintage, needs to be matured for 15 months or more. The cheese is kept at a constant temperature, often requiring special facilities. As with other hard cheese varieties produced worldwide, caves provide an ideal environment for maturing cheese; still, today, some cheddar is matured in the caves at Wookey Hole and Cheddar Gorge. Additionally, some versions of cheddar are smoked.

Character
The ideal quality of the original Somerset cheddar was described by Joseph Harding in 1864 as "close and firm in texture, yet mellow in character or quality; it is rich with a tendency to melt in the mouth, the flavour full and fine, approaching to that of a hazelnut".

Cheddar made in the classical way tends to have a sharp, pungent flavour, often slightly earthy. The "sharpness" of cheddar is associated with the levels of bitter peptides in the cheese. This bitterness has been found to be significant to the overall perception of the aged cheddar flavour. The texture is firm, with farmhouse traditional cheddar being slightly crumbly; it should also, if mature, contain large cheese crystals consisting of calcium lactate – often precipitated when matured for times longer than six months.

Cheddar can be a deep to pale yellow (off-white) colour, or a yellow-orange colour when certain plant extracts are added, such as beet juice. One commonly used spice is annatto, extracted from seeds of the tropical achiote tree. Originally added to simulate the colour of high-quality milk from grass-fed Jersey and Guernsey cows, annatto may also impart a sweet, nutty flavour. The largest producer of cheddar cheese in the United States, Kraft, uses a combination of annatto and oleoresin paprika, an extract of the lipophilic (oily) portion of paprika.

Cheddar was sometimes (and still can be found) packaged in black wax, but was more commonly packaged in larded cloth, which was impermeable to contaminants, but still allowed the cheese to "breathe".

Original-cheddar designation
The Slow Food Movement has created a cheddar presidium, arguing that only three cheeses should be called "original cheddar". Their specifications, which go further than the "West Country Farmhouse Cheddar" PDO, require that cheddar be made in Somerset and with traditional methods, such as using raw milk, traditional animal rennet, and a cloth wrapping.

International production
The "cheddar cheese" name is used internationally; its name does not have a protected designation of origin, but the use of the name "West Country Farmhouse Cheddar" does. In addition to the United Kingdom, cheddar is also made in Australia, Argentina, Belgium, Canada, Germany, Ireland, the Netherlands, New Zealand, South Africa, Sweden, Finland, Uruguay and the United States. Cheddars can be either industrial or artisan cheeses. The flavour, colour, and quality of industrial cheese varies significantly, and food packaging will usually indicate a strength, such as mild, medium, strong, tasty, sharp, extra sharp, mature, old, or vintage; this may indicate the maturation period, or food additives used to enhance the flavour. Artisan varieties develop strong and diverse flavours over time.

Australia
As of 2013, cheddar accounts for over 55% of the Australian cheese market, with average annual consumption around 7.5 kg per person. Cheddar is so commonly found that the name is rarely used: instead, cheddar is sold by strength alone as e.g. "mild", "tasty" or "sharp".

Canada
Following a wheat midge outbreak in Canada in the mid-19th century, farmers in Ontario began to convert to dairy farming in large numbers, and cheddar cheese became their main exportable product, even being exported to England. By the turn of the 20th century, 1,242 cheddar factories were in Ontario, and cheddar had become Canada's second-largest export after timber. Cheddar exports totalled 234000000 lbs in 1904, but by 2012, Canada was a net importer of cheese. James L. Kraft grew up on a dairy farm in Ontario, before moving to Chicago. According to the writer Sarah Champman, "Although we cannot wholly lay the decline of cheese craft in Canada at the feet of James Lewis Kraft, it did correspond with the rise of Kraft’s processed cheese empire." Most Canadian cheddar is produced in the provinces of Québec (40.8%) and Ontario (36%), though other provinces produce some and some smaller artisanal producers exist. The annual production is 120,000 tons. It is aged a minimum of three months, but much of it is held for much longer, up to 10 years.

Canadian cheddar cheese soup is a featured dish at the Canada Pavilion at Epcot in Walt Disney World.

Percentage of butterfat or milk fat must be labelled by the words milk fat or abbreviations B.F. or M.F.

New Zealand
Most of the cheddar produced in New Zealand is factory-made, although some are handmade by artisan cheesemakers. Factory-made cheddar is generally sold relatively young within New Zealand, but the Anchor dairy company ships New Zealand cheddars to the UK, where the blocks mature for another year or so.

United Kingdom
Only one producer of the cheese is now based in the village of Cheddar, the Cheddar Gorge Cheese Co. The name "cheddar" is not protected under European Union or UK law, though the name "West Country Farmhouse Cheddar" has an EU and (following Brexit) a UK protected designation of origin (PDO) registration, and may only be produced in Somerset, Devon, Dorset and Cornwall, using milk sourced from those counties. Cheddar is usually sold as mild, medium, mature, extra mature or vintage. Cheddar produced in Orkney is registered as an EU protected geographical indication under the name "Orkney Scottish Island Cheddar". This protection highlights the use of traditional methods, passed down through generations since 1946 and its uniqueness in comparison to other cheddar cheeses. "West Country Farmhouse Cheddar" is protected outside the UK and the EU as a Geographical Indication also in China, Georgia, Iceland, Japan, Moldova, Montenegro, Norway, Serbia, Switzerland and Ukraine.

Furthermore, a Protected Geographical Indication (PGI) was registered for Orkney Scottish Island Cheddar in 2013 in the EU, which also applies under UK law. It is protected as a geographical indication in Iceland, Montenegro, Norway and Serbia.

United States
The state of Wisconsin produces the most cheddar cheese in the United States; other centres of production include California, Idaho, New York, Vermont, Oregon, Texas, and Oklahoma. It is sold in several varieties, namely mild, medium, sharp, extra sharp, New York style, white, and Vermont. New York–style cheddar is particularly sharp/acidic, but tends to be somewhat softer than the milder-tasting varieties. Cheddar that does not contain annatto is frequently labelled "white cheddar" or "Vermont cheddar", regardless of whether it was actually produced there. Vermont's three creameries produce cheddar cheeses – Cabot Creamery, which produces the 16-month-old "Private Stock Cheddar"; the Grafton Village Cheese Company; and Shelburne Farms.

Some processed cheeses or "cheese foods" are called "cheddar flavoured". Examples include Easy Cheese, a cheese-food packaged in a pressurised spray can; also, as packs of square, sliced, individually-wrapped "process cheese", which is sometimes also pasteurised.

Cheddar is one of several products used by the United States Department of Agriculture to track the status of America's overall dairy industry; reports are issued weekly detailing prices and production quantities.

Records
U.S. President Andrew Jackson once held an open house party at the White House at which he served a 1400 lb block of cheddar. The White House is said to have smelled of cheese for weeks.

A cheese of 7000 lb was produced in Ingersoll, Ontario, in 1866 and exhibited in New York and Britain; it was described in the poem "Ode on the Mammoth Cheese Weighing over 7,000 Pounds" by Canadian poet James McIntyre.

In 1893, farmers from the town of Perth, Ontario, produced the "mammoth cheese", which weighed 22000 lb for the Chicago World's Fair. It was to be exhibited at the Canadian display, but the mammoth cheese fell through the floor and was placed on a reinforced concrete floor in the Agricultural Building. It received the most journalistic attention at the fair and was awarded the bronze medal. A larger, Wisconsin cheese of 34591 lb was made for the 1964 New York World's Fair. A cheese this size would use the equivalent of the daily milk production of 16,000 cows.

Oregon members of the Federation of American Cheese-makers created the largest cheddar in 1989. The cheese weighed 56850 lb.

In 2012, Wisconsin cheese shop owner Edward Zahn discovered and sold a batch of unintentionally aged cheddar up to 40 years old, possibly "the oldest collection of cheese ever assembled and sold to the public". The old cheese has extensive crystallization on the outside and is "creamier and overwhelmingly sharp" on the inside.

title: Cheddar cheese


text:
Classical order

An order in architecture is a certain assemblage of parts subject to uniform established proportions, regulated by the office that each part has to perform. Coming down to the present from Ancient Greek and Ancient Roman civilization, the architectural orders are the styles of classical architecture, each distinguished by its proportions and characteristic profiles and details, and most readily recognizable by the type of column employed. The three orders of architecture—the Doric, Ionic, and Corinthian—originated in Greece. To these the Romans added, in practice if not in name, the Tuscan, which they made simpler than Doric, and the Composite, which was more ornamental than the Corinthian. The architectural order of a classical building is akin to the mode or key of classical music; the grammar or rhetoric of a written composition. It is established by certain modules like the intervals of music, and it raises certain expectations in an audience attuned to its language.

Whereas the orders were essentially structural in Ancient Greek architecture, which made little use of the arch until its late period, in Roman architecture where the arch was often dominant, the orders became increasingly decorative elements except in porticos and similar uses. Columns shrank into half-columns emerging from walls or turned into pilasters. This treatment continued after the conscious and "correct" use of the orders, initially following exclusively Roman models, returned in the Italian Renaissance. Greek Revival architecture, inspired by increasing knowledge of Greek originals, returned to more authentic models, including ones from relatively early periods.

Elements
Each style has distinctive capitals at the top of columns and horizontal entablatures which it supports, while the rest of the building does not in itself vary between the orders. The column shaft and base also varies with the order, and is sometimes articulated with vertical concave grooves known as fluting. The shaft is wider at the bottom than at the top, because its entasis, beginning a third of the way up, imperceptibly makes the column slightly more slender at the top, although some Doric columns, especially early Greek ones, are visibly "flared", with straight profiles that narrow going up the shaft.

The capital rests on the shaft. It has a load-bearing function, which concentrates the weight of the entablature on the supportive column, but it primarily serves an aesthetic purpose. The necking is the continuation of the shaft, but is visually separated by one or many grooves. The echinus lies atop the necking. It is a circular block that bulges outwards towards the top to support the abacus, which is a square or shaped block that in turn supports the entablature. The entablature consists of three horizontal layers, all of which are visually separated from each other using moldings or bands. In Roman and post-Renaissance work, the entablature may be carried from column to column in the form of an arch that springs from the column that bears its weight, retaining its divisions and sculptural enrichment, if any. There are names for all the many parts of the orders.

Measurement

The heights of columns are calculated in terms of a ratio between the diameter of the shaft at its base and the height of the column. A Doric column can be described as seven diameters high, an Ionic column as eight diameters high, and a Corinthian column nine diameters high, although the actual ratios used vary considerably in both ancient and revived examples, but still keeping to the trend of increasing slimness between the orders. Sometimes this is phrased as "lower diameters high", to establish which part of the shaft has been measured.

Greek orders
There are three distinct orders in Ancient Greek architecture: Doric, Ionic, and Corinthian. These three were adopted by the Romans, who modified their capitals. The Roman adoption of the Greek orders took place in the 1st century BC. The three ancient Greek orders have since been consistently used in European Neoclassical architecture.

Sometimes the Doric order is considered the earliest order, but there is no evidence to support this. Rather, the Doric and Ionic orders seem to have appeared at around the same time, the Ionic in eastern Greece and the Doric in the west and mainland.

Both the Doric and the Ionic order appear to have originated in wood. The Temple of Hera in Olympia is the oldest well-preserved temple of Doric architecture. It was built just after 600 BC. The Doric order later spread across Greece and into Sicily, where it was the chief order for monumental architecture for 800 years. Early Greeks were no doubt aware of the use of stone columns with bases and capitals in ancient Egyptian architecture, and that of other Near Eastern cultures, although there they were mostly used in interiors, rather than as a dominant feature of all or part of exteriors, in the Greek style.

Doric order
The Doric order originated on the mainland and western Greece. It is the simplest of the orders, characterized by short, organized, heavy columns with plain, round capitals (tops) and no base. With a height that is only four to eight times its diameter, the columns are the most squat of all orders. The shaft of the Doric order is channeled with 20 flutes. The capital consists of a necking or annulet, which is a simple ring. The echinus is convex, or circular cushion like stone, and the abacus is a square slab of stone.

Above the capital is a square abacus connecting the capital to the entablature. The entablature is divided into three horizontal registers, the lower part of which is either smooth or divided by horizontal lines. The upper half is distinctive for the Doric order. The frieze of the Doric entablature is divided into triglyphs and metopes. A triglyph is a unit consisting of three vertical bands which are separated by grooves. Metopes are the plain or carved reliefs between two triglyphs.

The Greek forms of the Doric order come without an individual base. They instead are placed directly on the stylobate. Later forms, however, came with the conventional base consisting of a plinth and a torus. The Roman versions of the Doric order have smaller proportions. As a result, they appear lighter than the Greek orders.

Ionic order
The Ionic order came from eastern Greece, where its origins are entwined with the similar but little known Aeolic order. It is distinguished by slender, fluted pillars with a large base and two opposed volutes (also called "scrolls") in the echinus of the capital. The echinus itself is decorated with an egg-and-dart motif. The Ionic shaft comes with four more flutes than the Doric counterpart (totalling 24). The Ionic base has two convex moldings called tori, which are separated by a scotia.

The Ionic order is also marked by an entasis, a curved tapering in the column shaft. A column of the Ionic order is nine times more tall than its lower diameter. The shaft itself is eight diameters high. The architrave of the entablature commonly consists of three stepped bands (fasciae). The frieze comes without the Doric triglyph and metope. The frieze sometimes comes with a continuous ornament such as carved figures instead.

Corinthian order
The Corinthian order is the most elaborated of the Greek orders, characterized by a slender fluted column having an ornate capital decorated with two rows of acanthus leaves and four scrolls. The shaft of the Corinthian order has 24 flutes. The column is commonly ten diameters high.

The Roman writer Vitruvius credited the invention of the Corinthian order to Callimachus, a Greek sculptor of the 5th century BC. The oldest known building built according to this order is the Choragic Monument of Lysicrates in Athens, constructed from 335 to 334 BC. The Corinthian order was raised to rank by the writings of Vitruvius in the 1st century BC.

Roman orders
The Romans adapted all the Greek orders and also developed two orders of their own, basically modifications of Greek orders. However, it was not until the Renaissance that these were named and formalized as the Tuscan and Composite, respectively the plainest and most ornate of the orders. The Romans also invented the Superposed order. A superposed order is when successive stories of a building have different orders. The heaviest orders were at the bottom, whilst the lightest came at the top. This means that the Doric order was the order of the ground floor, the Ionic order was used for the middle story, while the Corinthian or the Composite order was used for the top story.

The Giant order was invented by architects in the Renaissance. The Giant order is characterized by columns that extend the height of two or more stories.

Tuscan order

The Tuscan order has a very plain design, with a plain shaft, and a simple capital, base, and frieze. It is a simplified adaptation of the Greeks' Doric order. The Tuscan order is characterized by an unfluted shaft and a capital that consists of only an echinus and an abacus. In proportions it is similar to the Doric order, but overall it is significantly plainer. The column is normally seven diameters high. Compared to the other orders, the Tuscan order looks the most solid.

Composite order
The Composite order is a mixed order, combining the volutes of the Ionic with the leaves of the Corinthian order. Until the Renaissance it was not ranked as a separate order. Instead it was considered as a late Roman form of the Corinthian order. The column of the Composite order is typically ten diameters high.

Historical development
The Renaissance period saw renewed interest in the literary sources of the ancient cultures of Greece and Rome, and the fertile development of a new architecture based on classical principles. The treatise De architectura by Roman theoretician, architect and engineer Vitruvius, is the only architectural writing that survived from Antiquity. Effectively rediscovered in the 15th century, Vitruvius came to be regarded as the ultimate authority on architecture. However, in his text the word order is not to be found. To describe the four species of columns (he only mentions: Tuscan, Doric, Ionic and Corinthian) he uses, in fact, various words such as: genus (gender), mos (habit, fashion, manner), opera (work).

The term order, as well as the idea of redefining the canon started circulating in Rome, at the beginning of the 16th century, probably during the studies of Vitruvius' text conducted and shared by Peruzzi, Raphael, and Sangallo. Ever since, the definition of the canon has been a collective endeavor that involved several generations of European architects, from Renaissance and Baroque periods, basing their theories both on the study of Vitruvius' writings and the observation of Roman ruins (the Greek ruins became available only after Greek Independence, 1821–1823). What was added were rules for the use of the Architectural Orders, and the exact proportions of them in minute detail. Commentary on the appropriateness of the orders for temples devoted to particular deities (Vitruvius I.2.5) were elaborated by Renaissance theorists, with Doric characterized as bold and manly, Ionic as matronly, and Corinthian as maidenly.

Vignola defining the concept of "order"

Following the examples of Vitruvius and the five books of the Regole generali di architettura sopra le cinque maniere de gli edifici by Sebastiano Serlio published from 1537 onwards, Giacomo Barozzi da Vignola produced an architecture rule book that was not only more practical than the previous two treatises, but also was systematically and consistently adopting, for the first time, the term 'order' to define each of the five different species of columns inherited from antiquity. A first publication of the various plates, as separate sheets, appeared in Rome in 1562, with the title: Regola delli cinque ordini d'architettura ("Canon of the Five Orders of Architecture"). As David Watkin has pointed out, Vignola's book "was to have an astonishing publishing history of over 500 editions in 400 years in ten languages, Italian, Dutch, English, Flemish, French, German, Portuguese, Russian, Spanish, Swedish, during which it became perhaps the most influential book of all times".

The book consisted simply of an introduction followed by 32 annotated plates, highlighting the proportional system with all the minute details of the Five Architectural Orders. According to Christof Thoenes, the main expert of Renaissance architectural treatises, "in accordance with Vitruvius's example, Vignola chose a "module" equal to a half-diameter which is the base of the system. All the other measurements are expressed in fractions or in multiples of this module. The result is an arithmetical model, and with its help each order, harmoniously proportioned, can easily be adapted to any given height, of a façade or an interior. From this point of view, Vignola's Regola is a remarkable intellectual achievement".

In America, The American Builder's Companion, written in the early 19th century by the architect Asher Benjamin, influenced many builders in the eastern states, particularly those who developed what became known as the Federal style. The last American re-interpretation of Vignola's Regola, was edited in 1904 by William Robert Ware.

The break from the classical mode came first with the Gothic Revival architecture, then the development of modernism during the 19th century. The Bauhaus promoted pure functionalism, stripped of superfluous ornament, and that has become one of the defining characteristics of modern architecture. There are some exceptions. Postmodernism introduced an ironic use of the orders as a cultural reference, divorced from the strict rules of composition. On the other hand, a number of practitioners such as Quinlan Terry in England, and Michael Dwyer, Richard Sammons, and Duncan Stroik in the United States, continue the classical tradition, and use the classical orders in their work.

Nonce orders
Several orders, usually based upon the composite order and only varying in the design of the capitals, have been invented under the inspiration of specific occasions, but have not been used again. They are termed "nonce orders" by analogy to nonce words; several examples follow below.

These nonce orders all express the "speaking architecture" (architecture parlante) that was taught in the Paris courses, most explicitly by Étienne-Louis Boullée, in which sculptural details of classical architecture could be enlisted to speak symbolically, the better to express the purpose of the structure and enrich its visual meaning with specific appropriateness. This idea was taken up strongly in the training of Beaux-Arts architecture, c. 1875–1915.

French order
The Hall of Mirrors in the Palace of Versailles contains pilasters with bronze capitals in the "French order". Designed by Charles Le Brun, the capitals display the national emblems of the Kingdom of France: the royal sun between two Gallic roosters above a fleur-de-lis.

British orders
Robert Adam's brother James was in Rome in 1762, drawing antiquities under the direction of Clérisseau; he invented a "British order" and published an engraving of it. Its capital the heraldic lion and unicorn take the place of the Composite's volutes, a Byzantine or Romanesque conception, but expressed in terms of neoclassical realism. Adam's ink-and-wash rendering with red highlighting is at the Avery Library, Columbia University.

In 1789 George Dance invented an Ammonite order, a variant of Ionic, substituting volutes in the form of fossil ammonites for John Boydell's Shakespeare Gallery in Pall Mall, London.

An adaptation of the Corinthian order by William Donthorne that used turnip leaves and mangelwurzel is termed the Agricultural order.

Sir Edwin Lutyens, who from 1912 laid out New Delhi as the new seat of government for the British Empire in India, designed a Delhi order having a capital displaying a band of vertical ridges, and with bells hanging at each corner as a replacement for volutes. His design for the new city's central palace, Viceroy's House, now the Presidential residence Rashtrapati Bhavan, was a thorough integration of elements of Indian architecture into a building of classical forms and proportions, and made use of the order throughout. The Delhi Order reappears in some later Lutyens buildings including Campion Hall, Oxford.

American orders
In the United States Benjamin Latrobe, the architect of the Capitol building in Washington, DC, designed a series of botanical American orders. Most famous is the Corinthian order substituting ears of corn and their husks for the acanthus leaves, which was executed by Giuseppe Franzoni and used in the small domed vestibule of the Senate. Only this vestibule survived the Burning of Washington in 1814, nearly intact.

With peace restored, Latrobe designed an American order that substituted tobacco leaves for the acanthus, of which he sent a sketch to Thomas Jefferson in a letter, 5 November 1816. He was encouraged to send a model of it, which remains at Monticello. In the 1830s Alexander Jackson Davis admired it enough to make a drawing of it. In 1809 Latrobe invented a second American order, employing magnolia flowers constrained within the profile of classical mouldings, as his drawing demonstrates. It was intended for "the Upper Columns in the Gallery of the Entrance of the Chamber of the Senate".

title: Classical order


text:
Talk:Classical order

Wiki Education Foundation-supported course assignment
This article was the subject of a Wiki Education Foundation-supported course assignment, between 4 September 2019 and 28 December 2019. Further details are available on the course page. Student editor(s): Psloann.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 17:47, 16 January 2022 (UTC)

Untitled
Hmm. Named something else? But unless the word 'classical' and 'order' are in the new title (Orders of classical architecture or some such) it will be less useful.
 * Or perhaps Classical orders of architecture. --Stephen Gilbert

Fine with me. I'm perhaps too close to the issue, because the term 'architecture' seems redundant. There aren't 'classical orders' in any other fields - classical forms, classical types, classical literature. --MichaelTinkler

 * There are classical orders in mathematics. But they're pretty obscure, and I doubt anyone will be writing an article on them any time soon. --Zundark, 2001 Oct 29

Changed category
I changed the category from 'Ancient Roman architecture' to 'Architecture', since the classical orders were neither invented by nor limited to the Romans. - Burschik

Moved article
I've moved the article from classical orders to classical order, as we should use the singular in article titles.

 * "Classical orders" means the orders used in Classical architecture. "Classical order", a characteristic for example of the versification and poetic vision of John Milton, is quite something else. Ideological consistency is a point of dogma rather than a neutral array of ideas. Remember Thoreau's dictum!. --Wetman 22:20, 15 Feb 2005 (UTC)

 * "Classical orders" is the plural. A quick Google shows that the singular is used far more often to refer to the architectural orders than to the concept (although a link to an article on it would be most welcome!).  A search of Britannica, for instance, returns almost thirty uses of "classical order" (many more, incidentally, than the plural), all referring to the architectural concept. Warofdreams 12:22, 16 Feb 2005 (UTC)

 * I'm confident we agree that this article concerns the Classical orders. Individual Wikipedia articles treat the Doric order, the Ionic order, the Corinthian order, even the Tuscan order, and ought to treat the Composite order too. When one is discussing the classical orders as a concept, the natural entry (think of the reader before you think of Wikipedian consistency) is the "Classical orders". Those Google hits simnply reflect the obvious fact that most reference are to an order being used in a particular building, not to the concept, as discussed for example by John Summerson, The Classical Language of Architecture. It was an unnecessary move, that's all. No problem, as long as there's a redirect. --Wetman 17:10, 16 Feb 2005 (UTC)

This text has been lifted from another source without credit.
"From the first formation of society, order may be traced. When the rigor of seasons first obliged humans to contrive shelter from the inclemency of the weather, we learn that they first planted trees on end, and then laid others across to support a covering. The bands which connected those trees at top and bottom are said to have given rise to the idea of the base and capital of pillars; and from this simple hint originally proceeded the more improved art of architecure."

This text comes directly from Masonic handbooks. Further, it has nothing to do with classical orders qua architecture as it is and should be moved to a discussion of "classical architecture" in terms of later European symbolism. Failing that, it should be cited.
 * It's an idea fashionable ca 1760-1600. Let's can it. --Wetman 05:44, 8 May 2006 (UTC)

"Modern approaches"
This section is too uncomprehending in toto to select an example. Someone who understands what "modern/Modernist" implies and has read any modern architect's dictum on the orders could easily improve this section. --Wetman 05:03, 14 August 2006 (UTC)

 * I'm working on it, made a start but now reading up.
 * Also I don't believe the expression 'nonce order' has ever been used except in this article and I'm considering whether it needs to be replaced with a more neutral term. This isn't the right place to invent a new name, if that's what it is. ProfDEH (talk) 22:11, 23 November 2009 (UTC)

History and cultural/social analysis?
It would be useful to have more information on the history of how these orders originated and became established as canonical, and on any theories as to why they were/are so incredibly conservative. This is the most fascinating aspect of the phenomenon to many casual observers such as myself; these three (or five) styles became (literally) set in stone at the dawn of western civilisation and have persisted ever since with only very minor attempts at modification or expansion. How, and why? The article touches on this by mentioning ideas about how they represent an understood grammar for architectural expression etc., but it would be good to have more.. Spiridens (talk) 09:01, 22 November 2009 (UTC)

The five pillars of ancient Greece have lost their meaning
The five pillars of Ancient Greek society are lost! Originally they stood for the senate, the people, philosophy, science and law, with selected pillars added for favorite gods who were thought to protect these. The discussions of Ancient Greek philosophers centered on their proof. The conclusion was that when the five load bearing pillars of the roman pantheon fall the apocalypse will come. — Preceding unsigned comment added by 101.163.69.209 (talk) 08:27, 2 September 2014 (UTC)

The Five Orders as Masculine/Matronly/Maidenly
I am by no means an expert in the history of architecture, which is why I'm posting here rather than being bold and editing the page myself. The text of the article says that the various classical orders are associated with various gender aesthetics: Doric (&Tuscan) represents masculinity, Ionic represents matronly femininity, and Corinthian (&Composite) represents maidenly femininity. Good so far. That's certainly how I learned it.

The problem comes when I followed the citation, which doesn't mention gender at all. Perhaps it was edited away? I realize that gender these days is a hot button issue and an easy way to get permabanned now or someday (another good reason why I didn't just revise the section myself, and am not using my account). However, when I started googling around for a good citation to replace the one used, I couldn't find anything that I felt was a Reliable Source that discusses it. The closest thing to something with editorial review was an article that allegedly quoted Vitruvius... and also gave his recommendations for which saints were appropriate for which Order. Which is patently ridiculous since Vitruvius died before Christianity even existed. So no dice there. It's surprising that such a widely quoted idea isn't showing up, and I don't have a translation of Vitruvius handy. Ware's "The American Vignola" was a landmark work, but I just flipped through it, especially Part I where he discusses the Orders, and again no mention.

I'm 90% certain that the Renaissance theorists held these beliefs about classical architecture and gender roles, but I'm hoping someone who knows this stuff better than I can find support for it in the literature. So far, I've come up empty. (If there really isn't any basis; like, if it turned out to be a Victorian fabrication or urban legend, then it's still probably noteworthy enough to include in the article, carefully described and cited as such.) 71.204.40.56 (talk) 15:32, 31 May 2021 (UTC)
 * There's lots like this on google. Male & female go back to Vitruvius, & no doubt beyond; saints do not. Johnbod (talk) 16:07, 31 May 2021 (UTC)

title: Talk:Classical order


text:
Colin Kapp

Derek Ivor Colin Kapp (3 April 1928 – 3 August 2007), Known as Colin Kapp, was a British science fiction author best known for his stories about the Unorthodox Engineers.

As an electronic engineer, he began his career with Mullard Electronics then specialised in electroplating techniques, eventually becoming a freelance consultant engineer.

He was born in Southwark, south London, 3 April 1928 to John L. F. Kapp and Annie M.A. (née Towner).

Cageworld series

 * 1) Search for the sun! (1982) (also published as Cageworld)
 * 2) The Lost worlds of Cronus (1982)
 * 3) The Tyrant of Hades (1984)
 * 4) Star Search (1984)

Chaos series

 * The Patterns of Chaos (1972)
 * The Chaos Weapon (1977)

Standalone novels

 * The Dark Mind (1964) (also published as Transfinite Man)
 * The Wizard of Anharitte (1973)
 * The Survival Game (1976)
 * Manalone (1977)
 * The Ion War (1978)
 * The Timewinders (1980)

Unorthodox Engineers
Collected in The Unorthodox Engineers (1979)
 * "The Railways Up on Cannis" (1959)
 * "The Subways of Tazoo" (1964)
 * "The Pen and the Dark" (1966)
 * "Getaway from Getawehi" (1969)
 * "The Black Hole of Negrav" (1975)

Other stories

 * "Breaking Point" (1959)
 * "Survival Problem" (1959)
 * "Lambda I" (1962)
 * "The Night-Flame" (1964)
 * "Hunger Over Sweet Waters" (1965)
 * "Ambassador to Verdammt" (1967)
 * "The Imagination Trap" (1967)
 * "The Cloudbuilders" (1968)
 * "I Bring You Hands" (1968)
 * "Gottlos" (1969), notable for having (along with Keith Laumer's Bolo series) inspired Steve Jackson's classic game of 21st century tank warfare Ogre.
 * "The Teacher" (1969)
 * "Letter from an Unknown Genius" (1971)
 * "What the Thunder Said" (1972)
 * "Which Way Do I Go For Jericho?" (1972)
 * "The Old King's Answers" (1973)
 * "Crimescan" (1973)
 * "What The Thunder Said" (1973)
 * "Mephisto and the Ion Explorer" (1974)
 * "War of the Wastelife" (1974)
 * "Cassius and the Mind-Jaunt" (1975)
 * "Something in the City" (1984)
 * "An Alternative to Salt" (1986)

title: Colin Kapp