Talk:Sierpinski triangle

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Java Code[edit]

I just compiled and ran the java code presented on the page. I got the following error:

  • java SierpinskiTriangle
  • java.lang.NoSuchMethodError: main
  • Exception in thread "main"

- —Preceding unsigned comment added by Russoc4 (talkcontribs) 10:16, 8 April 2006

The program in the article is an applet, you can't run it with java.exe. You need to first create an HTML page (.html file) with the applet in it and view it either from a web browser or with appletviewer.exe. I think the code in the HTML page is:


<applet code="">


Zchenyu 16:03, 29 July 2007 (UTC)
Most Java IDEs have an option to run as an applet, also. There should hopefully be no need to make a webpage just to run it. - Rainwarrior 21:08, 29 July 2007 (UTC)
You can also run applets from the command line using the appletviewer(1) command, on *nix or Windows. - SamB (talk) 17:02, 23 June 2010 (UTC)


About the Pascal's triangle you are not wrong, but the complete "triange" looks like:

       .   .   .   .   .   .   .   .   .   .   .   .
       .   .   .   .   .   .   .   .   .   .   .   .
       .   .   .   .   .   .   .   .   .   .   .   .
       0   0   0   0   0   0   1   
 . . 0   0   0   0   0   0   1  -5  15 -35
       0   0   0   0   0   1  -4  10 -20  35 -56  84
 . . 0   0   0   0   0   1  -3   6 -10  15 -21  28 -36  45 -55 . . .
       0   0   0   0   1  -2   3  -4   5  -6   7  -8   9 -10
 . . 0   0   0   0   1  -1   1  -1   1  -1   1  -1   1  -1   1 . . .
       0   0   0   1   0   0   0   0   0   0   0   0   0   0
 . . 0   0   0   1   1   0   0   0   0   0   0   0   0   0   0 . . .
       0   0   1   2   1   0   0   0   0   0   0   0   0   0
 . . 0   0   1   3   3   1   0   0   0   0   0   0   0   0   0 . . .
       0   1   4   6   4   1   0   0   0   0   0   0   0   0
 . . 0   1   5  10  10   5   1   0   0   0   0   0   0   0   0 . . .
       1   6  15  20  15   6   1   0   0   0   0   0   0   0
 . . 1   7  21  35  35  21   7   1   0   0   0   0   0   0   0 . . . 

So you don't get:

        # #
       #   #
      # # # #
     #       #
    # #     # #
   #   #   #   #
  # # # # # # # #
 #               #
# #             # #

More you get like this:

               # # # #         # # # #
              #       #       #       #
             # #     # #     # #     # #
            #   #   #   #   #   #   #   #
           # # # # # # # # # # # # # # # #. . . . .
         1 1
        1 2 1
       1 3 3 1
      #       #
     # #     # #
    #   #   #   #
   # # # # # # # #
  .               .
 .                 .
.                   .

-- 10:58, 19 Mar 2004 (UTC)

3D Sierpinskis[edit]

Take a look at Sv:Sierpinskitriangel. There you can find my 3D versions of the triangle. I have created the tool used to render the images. It is based on a three dimensional iterated function system. My technology to get the shadows there is probably new to the fractal scene. Please feel free to copy the images to here if you like. // Solkoll 01:29, 31 Jan 2005 (UTC)

I think they should be also displayed on the english page.
I can only guess what is written on the page but I can see you have written that the Hausdorff dimension is equal exactly 2. I Believe it should also be said on the english page. Also, It is noteworthy that a bijection to a square i an edge-first parallel projection of the bounding tetrahedron.
Just out of curiousity, what shadow technology were you using? - Rainwarrior 18:33, 27 July 2006 (UTC)

Programcode to draw Sierpinski Triangles?[edit]

Would it not be appropriate to include some (recursive) programcode (e.g. a Java Applet) to draw Sierpinski Triangles? I have done so in the corresponding articles in the german and french language wikipedia. — Nol Aders 13:28, 24 November 2005 (UTC)

I wrote an implementation of the Random-Convergence Sierpinski algorithm in C#, it doesn't use recursion, but is longer than the Java one posted (but unlike the Java one, this one compiles fine), should I post it? W3bbo 22:05, 2 August 2006 (UTC)

I discovered while messing around with bitwise "and" in QBasic that the following gives you Sierpinski Triangles:

10 For a = 1 to <number of pixels>
20 For b = 1 to <number of pixels>
30 If a and b then
40 PSET(a,b)
50 End If
60 Next b
70 Next a (talk) 14:18, 14 July 2008 (UTC)

wrong redirect?[edit]

I get redirected here when searching for "Tetrix" what is a class of Grashoppers... Does that have a reason?

According to the section "Analogs in higher dimension", a tetrix is a three-dimensional analog of the Sierpinski triangle. However, I agree that this is confusing for people looking for the insect, so I wrote a page at Tetrix saying that the word has two meanings. Unfortunately, we do not seem to have any information on the insect. Anyway, thanks for bringing this to our attention. -- Jitse Niesen (talk) 13:41, 16 January 2006 (UTC)


About the "Construction" part, I think that this sentence might be misleading:

"Shrink the triangle by 1/2, make three copies, and position the three copies so that each triangle touches the two other triangles at a corner (image 2)."

To me, making 3 copies would mean that there are 4 in total (including the original shrinked one). May be it should be change to "make 2 copies" or something.


Good point, thanks. I changed it to "Shrink the triangle by 1/2, make two copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2)." -- Jitse Niesen (talk) 06:08, 3 April 2006 (UTC)

Also, shouldn't "Repeat step 2 with each of the smaller triangles (image 3 and so on)" instead just be "Repeat step 2 (image 3 and so on)"?

In other words, you don't need to repeat with each of the three triangles, but can repeat with the full triangle instead. It seems that approach is cleaner/simpler as it involves a simple loop instead of a recursion.

Tom Dibble 00:01, 15 September 2007 (UTC)

Gasket or Carpet[edit]

I believe that the following passage should be moved to the Sierpinski carpet.

Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional plane, is homeomorphic to a subset of the Sierpinski gasket. For curves that cannot be drawn on a 2D surface without self-intersections, the corresponding universal curve is the Menger sponge.

Well, let's see his articles:

Gasket: Sur une courbe dont tout point est une point de ramification. C. R. Acad. Sci. Paris 160(1915) 302-305

Carpet: Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée. (French) C. R. 162(1916), 629-632.

Please let me know if I am wrong. --Beaumont 17:25, 27 July 2006 (UTC)

That's right. It should be at Sierpinski carpet. I came across that topic in a book called "The Beauty of Fractals" which also demonstrated specifically why the carpet could do this, but the triangle could not. - Rainwarrior 18:31, 27 July 2006 (UTC)


While the picture of the Sierpinski tetrahederon is cool, why is it on this page? - —The preceding unsigned comment was added by (talkcontribs) 20:37, 27 October 2006 (UTC)

Arguably, nowadays the Sierpinski triangle often means the object defined and studied in a N-dimensional setting (some sources on demand). Not a bad idea to have it here. --Beaumont (@) 20:58, 27 October 2006 (UTC)
This page is about many forms of the sierpinski triangle, so any material about the sierpinski tetrahedron belongs here. - Rainwarrior 06:59, 28 October 2006 (UTC)
The pictured item labeled 'Sierpinski tetrahedron', however cool looking, actually has 5 sides (a rectangular pyramid) User:Sarujin 8:20 AM, 11 November 2006 (UTC)
It seems to say "pyramid" now on the picture, but I added a note to the text nearby that there is also a pyramid construction. - Rainwarrior 21:14, 29 July 2007 (UTC)


Doesa anyone have a series/sequence formula for the total number of triangles, as well as a formula for finding the area of a triangle at stage "n", And is there a way to do this without using the recursive or "iterated function" method?--AeomMai 22:24, 3 January 2007 (UTC) Once you learn it, it is REALLY easy!

area - where n is iteration number - 0 is a triangle B is area of original triangle A = B * (3/4)^n Every time you iterate, you are basically multiplying the area by 3 but you are increasing the size of the entire triangle by 4

       X      XOOOX
 1 -> 3/4 ->  9/16 -> 27/64 ... 

number of triangles (black) -> 3^n

number of triangles (white) -> 4^n - 3^n

total area -> 4^n VentusIgnis 16:24, 18 June 2007 (UTC)

Naming: Sierpiński versus Sierpinski[edit]

Apparently this is named after Wacław Sierpiński, so why doesn't the spelling reflect that accurately here, as it does for Sierpiński's constant and Sierpiński curve? Does this need fixing up? —DIV 03:40, 6 January 2007 (UTC)

See Talk:Sierpinski number. I haven't examined "Sierpinski triangle". PrimeHunter 12:10, 6 January 2007 (UTC)

Pen and Paper[edit]

I remember a math teacher teaching a class on probability or chaos or something; asking us for some pairs of random numbers, plotting the medians of each pair, and ending up with a sierpinski triangle... it was a very dramatic demonstration... can anyone recall and describe this process? I think it'd be a good inclusion for an encyclopedia article. 02:21, 28 July 2007 (UTC)

There is a "chaos game" described in the article which involves randomness and midpoints (you take a random point, and then find the midpoint between that one and one of three fixed points, then repeat the process). Does it describe what you are talking about? - Rainwarrior 03:42, 28 July 2007 (UTC)

Initial Description[edit]

The initial description of the method of construction DOESN'T describe why the middle hole doesn't get filled in. I, personally, don't understand why it doesn't get filled in. My ONLY experience of the Sierpinski Triangle is from THIS article. So maybe that says something. :-) Martin Packer 11:40, 9 November 2007 (UTC)

Maybe step 2 in Construction should be reformulated. Step 2 replaces each triangle with 3 triangles of half the height, positioned inside the original triangle so they touch eachother at a corner (but not at a side). This positioning is part of the definition and it leaves a hole in the middle. PrimeHunter 12:23, 9 November 2007 (UTC)

I think that's MUCH better. The only tweak one might do is to spoon-feed the reader as to WHY that leaves a hole in the middle. Martin Packer 12:53, 9 November 2007 (UTC)

I'm not sure what you refer to with WHY. Do you mean why Sierpinski studied that form? I don't know. Do you want an explanation saying that 1/2 height and 1/2 width gives 1/4 the area, so it takes 3 smaller triangles and 1 hole of the same size to fill out the original triangle? Do you want a geometric explanation why the hole not only has the same size but also the same triangle form (upside down) as the 3 triangles? PrimeHunter 14:07, 9 November 2007 (UTC)
Thanks PrimeHunter: I really mean "why the step leaves the hole in the middle". It IS, I would agree, slightly spoonfeeding but the holes ARE a major feature of the triangle. Martin Packer 19:39, 9 November 2007 (UTC) (who claims to be a reasonable cipher for a layperson in this topic)

Physical model with clay[edit]

I would like to propose that an appropriate external link might be to this page, which shows one way to model the Sierpinski triangle physically, using an iterative process with modeling clay. I wrote the link in question, so I will not add it to the main article myself. But perhaps other editors would agree that it could be a useful addition to the page. —Preceding unsigned comment added by Oskay (talkcontribs) 22:12, 10 December 2007 (UTC)

Image not aligned[edit]

The Sierpinski triangle image is not horizontally aligned, which breaks its symmetry. You can better see this at full size. —Preceding unsigned comment added by Alphachapmtl (talkcontribs) 17:12, 19 January 2008 (UTC)

for "Analogs in Higher Dimention"[edit]

I have an image of the "Sierpiński triangle-based Pyramid" as seen from above and one as seen in 22.5°x22.5°x0° 3D. Are those considered appropriate images?Supuhstar * § 18:16, 30 June 2008 (UTC)

Well, go ahead and post it[edit]

Title says it all. As long as it's not genuinely horrible or pornographic, we'd like to see it.

I have image of "higher dimension" Sierpisnki appearing in the Chaotic Scattering problem[edit]

This is problem from Physics of particle scattering. Also some POV code to reproduce the results. I consider this to be "higher dimension" (actually 3d) because you can look around it. Can somebody add this image at the appropriate place in the text? —Preceding unsigned comment added by Aquarin (talkcontribs) 08:30, 1 August 2008 (UTC)

dino (talk) 00:13, 2 July 2008 (UTC)

Analog or Analogue?[edit]

What's the proper spelling in the context of "Analogs in higher dimension"? I'd have said 'analogue'. Tophtucker (talk) 16:41, 8 April 2009 (UTC)

There's no context, the difference in the spelling of "analog" is dependent upon whether you're an American writer or a British one. "Analog" is considered the standard English spelling, while "analogue" is considered the "Commonwealth English" spelling. (reference: wiktionary) (I consider the American spelling "standard" because my perception is that American English is the more influential language on a global scale, and thus assume more people use it, if you disagree, kindly replace the word standard with the word American, because I don't care enough to argue about it here) (Astrocom (talk) 07:45, 29 June 2010 (UTC))

worth mentioning ?[edit]

I was just reading the Blood type article and noticed that the Red blood cell compatibility table and the Plasma compatibility table look like the second and first approximations of the sierpinski triangle (when rotated). Is this worth mentioning in the article, e.g in a notes/in nature/... section ? what do you think ? --George (talk) 05:37, 5 October 2009 (UTC)

I don't agree, the tables do not look like approximations to the Sierpinski triangle, more like a square with a diagonal. /Pontus (talk) 18:58, 5 October 2009 (UTC)

I am talking about the table not the chart. It looks like this:

✔ ✔
✔     ✔
✔✔   ✔ ✔
✔ ✔  ✔   ✔

and there is an simple proof that this is not a coincidence: if "+" had the value 1, "A" the value 2 and "B" the value 4, every blood type will have numerical value from 0 to 7 (2^3 possible values) and a square will have a tick if (donor AND recipient = donor), this rule will produce a sierpinski triangle for any table with width 2**n. --George (talk) 23:11, 5 October 2009 (UTC)

Ok, sorry, I didn't look at the tables, just the charts. I think one could express a generalization of this in the following way. Let Qn be the directed n-dimensional hypercube graph. (The vertices are 0-1-sequences of length n, and there is a directed edge from x to y if x and y differs in exactly one position, at which x is 1 and y is 0.) Then the 0-1-matrix representing the partial order defined by reachability between the vertices of Qn (or, equivalently, the adjacency matrix of the transitive closure of Qn, with loops added at all vertices), with rows and columns of the matrix ordered by lexicographical order of the vertices, equals the (n-1)-st approximation of the Sierpinski triangle (viewed as a 0-1-matrix). I haven't found a reference for this. /Pontus (talk) 19:30, 6 October 2009 (UTC)
I realized that it is easier to define the partial order using the binary AND as you did: xy if (x AND y) = x, where x and y are integers from 0 to 2n-1. /Pontus (talk) 20:29, 6 October 2009 (UTC)

I didn't find a reference either, even though this is not very hard to prove. --George (talk) 21:58, 6 October 2009 (UTC)


I don't know how to word this proof or if it will be considered original research so I'll leave it here for now. For each trait (A,B,RhD) the donor (top) can donate blood to the receiver (left) if he is trait-negative or if both are trait-positive. at first this is demonstrated on the A antigen trait, then the B antigen trait and the Rhd antigen trait. the recursion produces an approximation of the sierpinski triangle. --George (talk) 22:31, 16 October 2009 (UTC)


Are there solid references?

dino (talk) 15:35, 5 October 2009 (UTC)

are you talking to me ? I can't tell, you started a new section... --George (talk) 17:44, 5 October 2009 (UTC)

Infinite Jest[edit]

What does the novel Infinite Jest have to do with this? Why is it in the See Also section? (talk) 17:26, 15 October 2009 (UTC)

removed. - Altenmann >t 17:38, 15 October 2009 (UTC)

Original Research?[edit]

I have a small amount of Original Research on this subject, far too small to ever be published as anything but commentary on the topic, and then, who would publish commentary on the topic? What should I do with it? I'm not a professor or a grad student, I don't have a sub-domain on a .edu site, so I have no way of putting this out in anyway that would be considered a reliable source.

If anyone is interested, I can elaborate on my (very small) amount of knowledge to add to the topic. (Astrocom (talk) 07:41, 29 June 2010 (UTC))

cosmati mosaic[edit]

In regard to this [1], I'm not questioning the reliability of the source. I do think however that this is a sort of an interesting trivia fact that can be put into the article but doesn't necessarily belong in the lede. But it's not a big deal either way.Volunteer Marek (talk) 21:58, 28 February 2011 (UTC)

It belongs wherever it is that we assign credit for its "discovery" to Sierpinski. Right now that's in the lede. But I agree that we aren't following MOS:LEDE in that the lede contains information not elaborated on in the rest of the article — not just the Cosmati mosaics but Sierpinski's role. —David Eppstein (talk) 23:12, 28 February 2011 (UTC)
Well, the thing is that fractal patterns occur in nature quite often, and art is often based on nature (particularly in mosaics) so it's not surprising that these fractals can be found in artistic works from long ago. And that's interesting. But that doesn't mean that the Cosmati mosaic "discovered" the Sierpinski triangle in any kind of mathematical sense, anymore than some sea snails "discovered" the golden ratio.Volunteer Marek (talk) 23:23, 28 February 2011 (UTC)
So only mathematicians are capable of making discoveries, and everyone else is at the intellectual level of a sea snail? That seems a very strange perspective on intellectual progress. I'd advise you not to say so in front of any artists. —David Eppstein (talk) 23:57, 28 February 2011 (UTC)
So only mathematicians are capable of making discoveries, and everyone else is at the intellectual level of a sea snail? - That's right.Volunteer Marek (talk) 01:00, 1 March 2011 (UTC)
No, ok, seriously, though I do know some mathematicians who take that view, there is a difference between a "mathematical discovery" (defining and characterizing a set) and a "artistic discovery" (finding a cool looking pattern for your mosaic) even if we're talking about the same thing. And yes, it is mathematicians who mostly make the "mathematical discoveries".Volunteer Marek (talk) 01:04, 1 March 2011 (UTC)
Ok, but is this article only about the mathematics of the Sierpinski triangle? And if so, where is that other article that covers the pattern in all its other aspects? And in any case I don't think "discovery" is the right word for what happens when a mathematician analyzes things that were already known (but not analyzed) to other people. —David Eppstein (talk) 01:13, 1 March 2011 (UTC)

Additional citations[edit]

Why, what, where, and how does this article need additional citations for verification? Hyacinth (talk) 16:04, 28 October 2011 (UTC)

Most of the construction section, all of the properties section, and all of the analogues section (that is to say, most of the article) have no footnotes or other forms of referencing indicating which of the stated facts come from which source. The few sources we have are not of the highest quality — we should easily be able to get published books for most of this material, rather than self-published web sites, which are already duplicated in the external links section. And the existing external links section is too long — many of those links should be cut. —David Eppstein (talk) 16:17, 28 October 2011 (UTC)

Javascript code[edit]

Hello guys,

here's a javascript code i've just made for the fractal, fully working on the html5 canvas:

Just call fractal_iteration(); for each iteration you want to do. Hope someone may post it :)

Thanks! — Preceding unsigned comment added by (talk) 01:04, 21 April 2012 (UTC)

cellular automata[edit]

What is an automaton "12/1" ? I can't seem to find anything about it on the net. — Preceding unsigned comment added by (talk) 19:54, 11 July 2012 (UTC)

(Please use the "New section" button up there to start a new topic, rather than putting your question above everything else, despite its earth-shaking importance.)
Clicking the nearest blue word, I find at Conway's Game of Life#Variations on Life that it means a version of Life in which a living cell survives if it has 1 or 2 living neighbors, and a dead cell comes to life if it has exactly 1 living neighbor. —Tamfang (talk) 03:05, 12 July 2012 (UTC)

Other means of generating[edit]

I know of at least two other methods to generate this pattern that aren't on the page. The first is created by using a binary tree, and is related to Pascal's triangle:

  1. Start with a single point
  2. Branch down to the left and right by a fixed amount and create a new point at each end
  3. From each point, branch to the left and right, omitting any branches which would end up landing in the same place
  4. Repeat

This will ultimately lead to this pattern. You can see this demonstrated here (Starting at about 2:00)

Another similar method creates not one, but 6 Sierpinski triangles in a hexagonal arrangement:

  1. Start with a single equilateral triangle
  2. Add another triangle to each face of the first
  3. Add more triangles to each face of the last triangles, all the same size
  4. Continue this pattern, but do not generate triangles where two of them would overlap.

You'll end up with 3 triangles at 120 degree angles to each other which all start with your original triangle at the tip, and 3 more at 60 degrees to those in which your original triangle serves upside-down as the first of the gaps (your 3 second-tier triangles forming the tips). Don't have a picture or video for this one, but you can try it out yourself with pen and paper. Lurlock (talk) 04:29, 13 February 2014 (UTC)

Fractal Curve, SVG and JavaScript[edit]

I have generated a Sierpinski Triangle using a single line in a fractal curve. Demonstration of this method is on my JsFiddle:

Sierpinski Triangle approximated by a fractal curve.

I believe this should be part of the article, since it shows how the triangle can appear from a completely different algorithm.

See Sierpiński arrowhead curve. I agree, this should be in the article; I just added it. —David Eppstein (talk) 22:49, 19 March 2014 (UTC)

Thanks David Eppstein, I didn't know about the Sierpiński arrowhead curve until your edits! I just stumbled upon it while trying to make a dragon curve. Alancnet (talk) 16:54, 20 March 2014 (UTC)

Is there a non-iterative way to generate it?[edit]

Like, is there a formula where you input the X and Y of each of the corners, and an arbitrary X and Y, and it will say whether those coordinates are in one of the empty areas, or the probability that those coordinates will be in an empty area? --TiagoTiago (talk) 22:06, 17 January 2018 (UTC)

Yes; just like the Cantor set, one can give a description in terms of the expansion in a particular base. This is alluded to in the third paragraph of the section properties. A different way to formulate it is as follows: the Sierpinski triangle that you get by starting with the triangle with vertices (0, 0), (1, 0) and (0, 1) consists of all points (X, Y) of two real numbers in [0, 1) such that when X and Y are expressed in binary, they do not both have a 1 in the same position; or equivalently, that computing the sum X + Y involves no carries.
(The question about randomly choosing a point is answered in the same section: the probability that a randomly chosen point belongs to the removed regions is 1.) --JBL (talk) 01:06, 18 January 2018 (UTC)

Hm, I missed that somehow.
Ok, but aren't those two things contradicting each other? If the probability is 1, then doesn't that mean there are no pairs of binary numbers between 0.0 and 1.0 with that property? I mean, am I missing something, or for example 0.1 and 0.01 would actually be part of the triangle, and therefore have a probability of zero of belonging to a removed region?
Also, here's an additional related question: Is there a non-iterative way to calculate how many iterations of the remove the middle upside-down triangle of each filled triangle algorithm it would take to remove any given point? --TiagoTiago (talk) 09:02, 19 January 2018 (UTC)
No: in continuous probability, "probability 0" and "does not occur" are not synonymous. (For example, if you choose a random real number between 0 and 5, the probability that you choose the number pi is 0.) If you use the barycentric encoding, then I think a point with coordinates (X, Y, Z) gets removed in the nth stage if the earliest position at which the condition stated in the article fails is nth position after the decimal point. --JBL (talk) 12:49, 19 January 2018 (UTC)