Talk:Homothety/Archive 1
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| Archive 1 |
[Untitled]
Unicode has character ∻ ([1]) on this table: http://www.decodeunicode.org/w3.php?ucHex=2200 . Is this character actually in usage to denote "homothetic"? --Abdull 21:42, 28 May 2006 (UTC)
Self-contradictory introduction
The first few sentences are as follows:
"In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point A called the origin. The number c by which distances are multiplied is called the dilation factor or similitude ratio. Such a transformation is also called an enlargement.
More generally c can be negative; in that case it not only multiplies all distances by | c | , but also inverts all points with respect to the fixed point."
It is absurd to first state that c is the factor by which distances are multiplied (which implies that c is necessarily >= 0) and then go on to say that "c can be negative" -- which is utterly nonsensical, in that it completely removes any meaning to what the number c is.
Why not get it right the first time? And by the way, the phrase "more generally" in this case does nothing to mitigate the nonsense. The phrase "More generally, c can be negative . . ." makes no sense regardless, since the meaning of "c" is entirely unclear.Daqu 05:38, 14 April 2007 (UTC)
- The definition is clear as is. The definition contains two parts: (i) a usual meaning (when c is >=0) and (ii) a natural extension when c < 0. The action of c as described imply the structure of the space it acts on; such as metric, vector, etc. Definitions of concepts need not explicitly define every aspect of the universe they live in. Else, many concept would require infinitely long definitions which is not practical to human epistemic pursuits. —Preceding unsigned comment added by 130.22.48.2 (talk) 14:15, 19 September 2007 (UTC)
First two paragraphs still wrong
I just looked at the first paragraph again today and -- however it may have happened -- it's now considerably worse than before:
"In mathematics, a homothety (or homothecy or dilation) is a transformation of space which takes each line into a parallel line (in essence, a similarity that is similarly arranged). All dilatations form a group in either affine or Euclidean geometry. Typical examples of dilatations are translations, half-turns, and the identity transformation."
It's true that a homothety takes each line to a parallel line ... but this is true of every linear or affine transformation, almost all of which are not homotheties. The phrase "a similarity that is similarly arranged" is cute, but conveys nothing, since no one knows what "similarly arranged means but the writer.
(And although the incorrect word "dilation" is used in the first sentence, the correct word -- "dilatation" -- is what's used in the second sentence without previous mention.)
Then -- the second paragraph begins with the same error discussed above:
"In Euclidean geometry, when not a translation, there is a unique number c by which distances in the dilatation are multiplied. It is called the ratio of magnification or dilation factor or similitude ratio. Such a transformation can be called an enlargement. More generally c can be negative; in that case it not only multiplies all distances by | c | , but also inverts all points with respect to the fixed point."
This implies that a ratio of distances can be negative, by first defining the constant c to be the ratio of distances, and then in the next sentence saying that "more generally, c can be negative" -- which contradicts the first sentence. This leaves the meaning of the constant c unclear -- and in odd dimensions it is false that multiplication by a negative number is a homothety.
The number c is best left as how it is defined -- namely, the ratio of new distance to old distance when a homothety is applied. Then the example can be described as a 180-degree rotation along with (followed or preceded by) a uniform scaling. There is no point in emphasizing the multiplication by a negative number, especially since that fails to be a homothety in odd-dimensional Euclidean space, like the line or 3-space.Daqu (talk) 19:42, 12 January 2009 (UTC)