Beat 1 C of the butter until it becomes fluffy and lighter colored. add to list. Painting in Swing is controlled by the RepaintManager, it is it's responsibility to determine what and when to repaint the screen. A Sierpinski Triangle is created by starting with an equilateral triangle and then subdividing it into smaller equilateral triangles. Tattoo Styles. Task. We first prove the appearance of more general fractals when Pascal's triangle is considered modulo prime. + (1,0))) # make a recording of figure `f` with 300 frames record(f. Ignoring the middle triangle that you just created, apply the same procedure to. fractal sierpinski-triangle fractal-geometry. <Alt>-click the shape to raise the number of iterations. It is an HTML canvas where I draw the Sierpinski triangle with JavaScript. There are di erent ways to construct it, and one of them is by shrinking and duplication [7]. Closely related to the gasket is the Sierpinski carpet. png → File:Sierpinski triangle evolution. The Sierpinski triangle generates the same pattern as mod 2 of Pascal's triangle. fractal sierpinski-triangle fractal-geometry. Pick a starting point at random. Task. Do the same for the three largest equilateral triangles left in this one. Welcome to the r/Tattoos subreddit community. If its n value is not zero: Draw the triangle connecting the midpoints of the triangle. What is the second step? (Sierpinski Triangle) Make the triangle half its height and width. 47. Repeat step 2 for each of the remaining smaller triangles forever. Black and Grey tattoos. This essentially simulates the recursion iteratively. A tattoo like this will perfectly represent masculinity with a swift touch of wisdom and mysticism. The Sierpinski triangle (also with the original orthography Sierpiński), also called the Sierpinski gasket or Sierpinski sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. unexpected properties, of which the Sierpiński gasket is the most famous. If we start with a single square filled in (i. In fact, Pascal's triangle mod 2 can be viewed as a self similar structure of triangles within triangles, within triangles, etc. pyplot based on 3 dots (x,y) in 2D? For instance, to compose a Sierpinski triangle from polygons, and plot those polygons onto a figure: The Sierpinski Triangle is a beautiful and intricate fractal pattern that has captured the imagination of mathematicians and artists alike. /. Drawing a Sierpinski triangle by hand. The Sierpinski gasket is constructed iteratively and is the result of an infinite number of steps. We could use frag to create filled triangles, but we need to avoid z-fighting by adding a little bit of code to change the elevation of each ‘level’: TO sierpinski :size :level if :level > 0 [ pu setz 0 lower 0. But people with only one of those still have that tattoo thing with all three triangles. Get yourself a 3-sided die. As in Figure 4, we see that this point hops into one of the three next-smaller triangles, since these triangles represent all points that are half the distance to the three vertices from points in the largest removed triangle. This really helps bring out the contrast in the larger and smaller triangles. Note that (1) all of the sets Gj and Tjk are triangles contained in the Sierpinski gasket, and (2) we have not relabeled the triangle G, as it has already been counted (in the previous stage of the construction). Divide this large triangle into four new triangles by connecting the midpoint of each side. Math. A Sierpinski triangle (in black) looks like the following. The probably most well-known occurrence of the Sierpinski Triangle is as the odd entries of the Pascal triangle. The Sierpinski Triangle (according to wikipedia) is a fractal in the shape equilateral triangle, containing other replicated triangles within it. The Polish mathematician Wacław Sierpiński described the pattern in 1915, but it has appeared in Italian art since the 13th century. The Sierpinski triangle of order 4 should look like this: Related tasks. All orders are custom made and most ship worldwide within 24 hours. A Sierpinski triangle is a self-similar fractal described by Waclaw Sierpinski in 1915. ago. Sierpinski triangle/Graphical for graphics images of this pattern. Just see the Sierpinski Triangle below to find out how infinite it may look. The recursion should stop when n is 0. Wacław Sierpiński foi o primeiro matemático a pensar nas. 99. Sierpinski triangle evolution, Wikipedia. Tα = φαm ∘ ⋯ ∘φα2 ∘φα1(T) and Gα =φαm ∘ ⋯ ∘φα2 ∘φα1(G). Sierpinski triangle within a delta symbol + variable x. A Sierpinski triangle or Sierpinski triangle gasket is a fractal resulting from doing the following: [1] Start with an equilateral triangle. geometry sierpinski-triangle fractals sierpinski-carpet algorithms-and-data-structures nanyang-polytechnic. The ST fractal is obtained in the limit of high number of iterations. Dec 13, 2019 - Explore Melissa McCaskill's board "Sierpinski Triangle Quilt", followed by 239 people on Pinterest. Furthermore, the Sierpinski triangle has zero area: this can be. Add a comment. Divide that triangle into four equilateral. , Nm = 3m. Draw a new triangle by connecting the midpoints of the three sides of your original triangle. The Triforce has a little more meaning. One of our problems was to create a Sierpinski triangle in stage 1,2, and 3 and find the total area of. Produce an ASCII representation of a Sierpinski triangle of order N. This is the kind of shit TOOL would figure out and it would be Danny Carey's drum solo. Screen () self. Sierpinski (1882-1969), which requires the following steps for its construction: start with an equilateral triangle, indicated with. Shrink the triangle to half height, and put a copy in each of the three corners. It is a self similar structure that occurs at different levels of iterations, or magnifications. Figure 2 shows theThe Sierpinski triangle is a great illustration of this effect. It is created by “infinite repetition” of the following steps: (1) for every filled equilateral triangle, locate the midpoints of each side, (2) connect these midpoints to form a smaller triangle, and (3) remove that triangle. Each small section of the Sierpinski triangle looks like a miniature version of the whole thing. svg is a vector version of this file. Discover (and save!) your own Pins on PinterestSierpinski Triangle | Apr 20th 2018 | 512708. 5 . Baroque tattoos. A level-4 Sierpinski triangle obtained by shading the first 32 levels of a Pascal triangle white if the binomial coefficient is even and black otherwise by CMG Lee. An IFS and an attractor are de ned in Section 2. It is impossible to draw a point in the whitespace middle of the original 3 points because that is not halfway between any 2 existing points. fillPolygon (px, py, 3); g. Fine Line Tattoos Victoria BC. 5 . 5. For more information, see Help:SVG. My screenshot is below the code. /. Add a description, image, and links to the sierpinski-triangle topic page so that developers can more easily learn about it. This process is repeated over and over again with the resulting triangles to produce the Sierpinski triangle, as. class Sierpinski: def __init__ (self): self. The 3D Printed Sierpinski Pyramid. Next, there are three recursive calls, one for each of the new corner triangles we get when we connect the midpoints. The Binary Sierpinski Triangle sequence is the sequence of numbers whose binary representations give the rows of the Binary Sierpinski Triangle, which is given by starting with a 1 in an infinite row of zeroes, then repeatedly replacing every pair of bits with the xor of those bits, like so: f (0)= 1 =1 f (1)= 1 1 =3 f (2)= 1 0 1 =5 f (3)= 1 1. 3. Example of use:350 4 6. Fractals III: The Sierpinski Triangle The Sierpinski Triangle is a gure with many interesting properties which must be made in a step-by-step process; that process is outlined below. The procedure for drawing a Sierpinski triangle by hand is simple. 1. xvals, out(i). Default value 0. The answer to 2) is more complicated, because in any language writing graphics is more complicated, because the hardware changes. Shoulder Tattoos. midpoints of the existing triangle to make a new, downward-facing triangle. Set v n+1 = 1 / 2 (v n + p r n),. + (1,0))) # make a recording of figure `f`. add to list. Use the Sierpinski 1 macro to create a second iteration Sierpinski Triangle by clicking on each of the lines joining the midpoints. html This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. Randomly select any one of the three triangle points. Repeat step 2 for each of the remaining smaller triangles forever. When drawing a new point, you pick 2 points that already exist and draw a new point in the middle. Such a tattoo on the dorsal side of a man’s hand can really upgrade your looks. View License. You could make the argument that the middle portion of the initial triangle can accommodate a fourth triangle, but we are disallowing rotation, so that. Painless and easy to apply. I could not see the point in adding the extra load of VUE and wrote a native example. Explore. The Sierpinski Triangle is a fractal named after a Polish mathematician named Wacław Sierpinski, who is best known for his work in an area of math called set theory. 000 -> 0 001 -> 1 010 -> 0 011 -> 1 100 -> 1 101 -> 0 110 -> 1 111 -> 0. The result looks like a fractal called the Sierpinski triangle or Sierpinski gasket. It is not always easy to determine if a fractal is connected. Divide it into 4 smaller congruent triangle and remove the central triangle . File usage on other wikis. <p style=”margin: 0px; font-size: 12px; line-height: normal; font-family: Helvetica;”>Hi, I am trying to decide whether to get my first tattoo on my shoulder or. Example. For comparison, the colour of the outline of its background is green, yellow or purple for the coefficient modulo 3 being 0, 1 or 2, respectively. Victoria, BCSierpinski Tetrahedra. Sep 19, 2016 - Welcome to the r/Tattoos subreddit community. Command (aka. Next, roll a die. 22-Mar-2022The Sierpinski triangle is a fractal described by Waclaw Sierpinski in 1915. It could also be written as reduce t = t === (t ||| t) The command to produce the SVG output is sierpinski -o Sierpinski-Haskell. In the Tower of Hanoi puzzle, disks stacked on one peg are to be moved to another with. If the original triangle is an obtuse triangle, the largest value of iter is 12. The triangle should be in the bottom center of your window. Therefore my intuition leads me to believe it's topological dimension is 1 (as the topological dimension must be less than the Hausdorff dimension). But people with only one of those still have that tattoo thing with all three triangles. You create a pyramid from p with (pyramid p). Size of this PNG preview of this SVG file: 680 × 111 pixels. And here in Sierpinski triangles, I needed so many lines of code. The Sierpinski triangle fractal was first introduced in 1915 by Wacław Sierpiński. e. " You can create the Sierpinski Triangle (and very similar fractals) with surprisingly little code. Discover (and save!) your own Pins on PinterestApr 13, 2022 - This Pin was discovered by Wendy Thacker. Describe the procedure (recursion) to construct the Sierpinski triangle in your own words. Take a piece of paper. Thank you for this. The first step of creating a Sierpinski triangle is constructing a large equilateral triangle. The Sierpinski triangle (also with the original orthography Sierpiński), also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. This means that any changes to count will only be local to the method, and not change the count object which is passed into the method from the parent method. Fractal Properties of the Sierpinski Triangle 5. The guy is like wow it appears randomly. 3. Organic ink. This Sierpinski Triangle studio shows at least 17 different methods of drawing the Sierpinski Triangle. Complementary main of a plane continuum X is any component of the complement of X . → Print-friendly version. This course is intended f. Furthermore, it is the only such triangle other than the ordinary middle triangle. Pick any number inside Pascal’s triangle and look at the six numbers around it (that form alternating petals in the flowers drawn above). For a real Sierpinski triangle, this process must be repeated forever, so that there are infinitely many triangles that are infinitesimally small. Thus, the dimension of a Sierpinski triangle is log (3) / log (2) ≈ 1. If you want more layers of the Sierpinski triangle then you need to add another loop, within the loop scale the current point by sqrt (3) and add another random rv. ”. The first of plane figures in Sacred Geometry and one of the most significant. Produce an ASCII representation of a Sierpinski triangle of order N. 3 . Discover (and save!) your own Pins on PinterestThe Sierpiński triangle named after the Polish mathematician Wacław Sierpiński), is a fractal with a shape of an equilateral triangle. The Sierpinski triangle is a fractal described in 1915 by Waclaw Sierpinski. The Sierpinski triangle is another example of a fractal pattern like the H-tree pattern from Section 2. Tags Spiral Vase Mode Sierpinski Pyramid. Starting with a simple triangle, the first step, shown in the. ; Sierpinski carpetThe Hojo Clan’s “Mitsuuroko” (三つ鱗) But in Japan, where Zelda was created, things are a little different. Download Wolfram Notebook. The triangle self-assembled through halogen and hydrogen. Funny, cool, or just plain weird, you'll find the socks your feet deserve. ; Sierpinski carpetThe answer to 1) is obviously yes, since Fortran is a general programming language. 3 of the textbook. Posted by 8 years ago. ; Sierpinski carpetTask. The Sierpin´ski triangle [Fig. Explore math with our beautiful, free online graphing calculator. The height method and filledTriangle method seem to be working fine since the triangles are equilateral and they are correctly filled and being printed. But it is more than 1-dimensional because one can prove that it has places of “density”, in other. Art----2. This packet leads students through the first few stages of two fractals - the Sierpinski Carpet and Sierpinski Triangle. First we will begin with the process of repeated removal, and an exploration of the Sierpinski Triangle. For example, the sub-triangle at prefix (x= exttt{132}) is obtained by taking the first tridrant of the base triangle, followed by the third tridrant within this sub. 4. The kth Sierpinski triangle is a triangle whose interior is sub-divided as follows: Take the three mid-points of the sides of the triangle. To review, open the file in an editor that reveals hidden Unicode characters. Divide: To create the Sierpinski triangle, we start with a. Below is the program to. 河內圖. 2 . The pattern was described by Polish mathematician Waclaw Sierpinski in 1915, but has appeared in Italian art since the 13th century. Give examples to show the self-similarity of the Sierpinski triangle. An algorithm for obtaining arbitarily close approximations to the Sierpinski triangle is as follows: Start with any triangle in a plane. Each students makes his/her own fractal triangle composed of smaller and smaller triangles. wikipedia. normal; font-family: Helvetica;”>Hi, I am trying to decide whether to get my first tattoo on my shoulder or lower on my arm. ; Sierpinski carpetHow to generate a Sierpinski triangle in code:Choose a random point in a triangle, then successively:Draw a dot at that pointChoose one of the vertices of th. See recursion stock video clips. Example. Start with a single large triangle. Here is a Sierpinski triangle where the three sub-triangles have each been drawn in a different colour: Here is the interesting fact - each of the three different coloured triangles is an exact copy of the original triangle. Produce an ASCII representation of a Sierpinski triangle of order N. Viewed 586 times. forward(size) t. In 10-20 steps, the point off S is practically indistinguishable from a similar point on S. The second iteration looks like this and has an area of 9/16units²: At each iteration, we note that the area of the “triangle” is 3/4 of the previous. Hate to burst the bubble but he’s following rules. Sale price $54. 585. Here's the most concise way I was able to come up with. Triangle is one of the most powerful and universal symbols. 2 width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). 19 November 2015. The Sierpinski pyramid is the 3-dimensional version of the Sierpinski triangle, named after the Polish mathematician Waclaw Sierpinski. The Sierpin´ski triangle [Fig. The triangle should be in the bottom center of your window. The Sierpiński triangles have been known for more than one hundred years, but only recently discrete shape-persistent low-generation (mainly ST-1) fractal supramolecules have been realized. The most conceptually simple way of generating the Sierpinski Triangle is to begin with a (usually, but not necessarily, equilateral) triangle (first figure below). File usage on Commons. Sierpiński Triangle - a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Repeat this process for the unshaded triangles in stage 1 to get stage 2. Therefore, it also generates a self-similar fractal, and we call improve wording it a Sierpinski pedal triangle (SPT)´. Sierpinski triangle/Graphical for graphics images of this pattern. 3. Overview. A Sierpinski triangle is a self-similar fractal described by Waclaw Sierpinski in 1915. Click on a date/time to view the file as it appeared at that time. Sierpinski triangle within a delta symbol + variable x. Therefore my intuition leads me to believe it's topological dimension is 1 (as the topological dimension must be less than the Hausdorff dimension). The chaos game is played as follows. When you call sierpinski recursively, you should pass it x instead of x+x/2. The central triangle is removed and each of the other three treated as the original was, and so on, creating an infinite regression in a finite space. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Painless and easy to apply. Next, students cut out their own triangle. "Write an algorithm and a Python implementation to draw a Sierpinski triangle. Example. For 2D point inputs, just padd the point Z coordinate with a zero (see example #2) To benefit from the file documentation attached, be sure to download the file, not to just copy and paste it. Find a midpoint of each side of the triangle below. The base state for this fractal is a single triangle. def drawSurroundingTriangles(startx : Double, starty : Double, width. However, since it's area is 0 this makes me believe. The function I used was: def sierpinski (screen, x, y, size, MinSize): if size <= MinSize: #creating a new triangle object T = triangle (x, y, size, white) #drawing the triangle to screen T. Triangles. draw (screen) #adding the triangle to the array Triangle. This is the order zero triangle. Produce an ASCII representation of a Sierpinski triangle of order N. The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis. Your function should print n and size, then recursively call itself three times with the arguments n - 1 and size / 2. The Tower of Hanoi: Where maths meets psychology. Sierpinski-like triangles can also be constructed on isosceles or scalene triangles. Specifying the length and depth in the constructor might allow you to have more control, by changing values at one place, you modify it all. Looks great. fractales. The Polish mathematician Wacław Sierpiński described the pattern in 1915, but it has appeared in Italian art since the 13th century. And so forth. 585, which follows from solving 2d = 3 for d. It is a three-dimensional generalization of the one-dimensional Cantor set and two. The Mandelbrot Set. In this paper we consider a quantum version of Pascal's triangle. In that case replace drawPolygon with fillPolygon and the triangles will be filled in. wikipedia. Example. Updated Jun 16, 2019. 1] is geometrically defined as follows. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in the Sierpinski triangle. Sierpinski fractal. 43). Produce an ASCII representation of a Sierpinski triangle of order N. Example. Searching for ‘mocap suit sierpinski’ leads via a post on Reddit, Why. Follow. Triângulo de Sierpinski. | Inkbox™ | Semi-Permanent Tattoos. : You are free: to share – to copy, distribute and transmit the work; to remix – to adapt the work; Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. Visual Studio 2017Ignoring the middle triangle that you just created, apply the same procedure to each of the three corner triangles. File:Sierpinski triangle evolution. append (T) else: #halving the size and then recalling this function size = int (size / 2. Based on the equilateral triangle, the Sierpinski triangle is a fractal - a geometric construction made up of patterns that are self-similar - smaller replicas of the larger version. As example I use the Sierpinski Triangle (Sierpinski Curve). An abstract 3D rendering background. Task. The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Waclaw Sierpinski. The following image is not an image. bgcolor('lightblue') alex=turtle. Instead of removing the central third of a triangle, the central square piece is removed from a square sliced into thirds horizontally and vertically. Once downloaded, typewrite 'doc Sierpinski_triangle' or 'help Sierpinski_triangle' in Matlab console for support. The Sierpiński sieve is a fractal described by Sierpiński in 1915 and appearing in Italian art from the 13th century (Wolfram 2002, p. Don't do that here. This process is shown in Figure 3. It should be used in place of this PNG file when not inferior. The Sierpinski triangle is what's known as a fractal: an object that is infinitely similar to itself. The Sierpinski triangle activity illustrates the fundamental principles of fractals – how a pattern can repeat again and again at different scales and how this complex shape can be formed by simple repetition. The Mandelbrot Set. Then we use the midpoints of each side as the vertices of a new triangle, which we then remove from the original. I happened to see a tweet from @LASTEXITshirts celebrating the birthday of actor Andy Serkis (56 today), and I recognised the pattern on the suit he’s wearing. A stop criteria. The Sierpinski Triangle is a thing of mesmerising beauty to the mathematically minded and all those who appreciate the concept of infinity. 6. I will give a short description of the algorithm which is used to draw the Sierpinski curve and show how to use the combination of JavaScript and the HTML5 canvas element. Sierpinski Triangle, Wall Tapestry. [1] He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis ), number theory, theory of functions, and topology. 3. e. There is a similar method that is. tol: A stopping creiteria to draw the sub-SPT. File. The Sierpinski triangle has Hausdorff dimension log(3)/log(2) ≈ 1. From $43. O Triângulo de Sierpinski - também chamado de Junta de Sierpinski - é uma figura geométrica obtida através de um processo recursivo. If its n value is not zero: Draw the triangle connecting the midpoints of the triangle. By Morgan Gatekeeper: Tempe, AZ. import turtle def draw_triangle (some_turtle): #This for loop will create - Outer Triangle some_turtle. Um dos fractais que vimos no capítulo anterior foi o triângulo de Sierpinski, que recebeu o nome do matemático polonês Wacław Sierpiński. Nine different waysPlotting the good old Sierpinski triangle. Unique Sierpinski Triangle Posters designed and sold by artists. Ignoring the middle triangle that you just created, apply the same procedure to. 2. . The Sierpiński triangle is a modified version where a. This is because, in this program, you are using the bottom right triangle vertex as the primary. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. The guy is like wow it appears randomly. The Sierpinski triangle can be described as the result of an "iterated function system. Each students makes his/her own fractal triangle composed of smaller and smaller triangles. imgur. Repeat steps 2 and 3 for each remaining triangle, removing the middle triangle each time. Pascal's triangle is a well-known triangular array of numbers and when these numbers are plotted modulo 2, a fractal known as the Sierpinski triangle appears. From Wikimedia Commons, the free media repository. Available in a range of colours and styles for men, women, and everyone. The recursive structure. This creates a struct of length 3^n, each entry of which contains the coordinates of one of the small triangles in the sierpinski triangle. Produce an ASCII representation of a Sierpinski triangle of order N. Art. Stage 0:Begin with an equilateral triangle with area 1, call this stage 0, or S 0. ; Sierpinski carpetIf one takes a point and applies each of the transformations d A, d B, and d C to it randomly, the resulting points will be dense in the Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations to it:. 3 of the textbook. Knowing how to create repeating and growing patterns, understand relations, and functions doesn’t necessary mean that students won’t make mistakes. When was Sierpinski triangle invented? 1915 The. Self-similar means when you zoom in on a part of the pattern, you get a perfectly identical copy of the original. Sep 19, 2016 - Welcome to the r/Tattoos subreddit community. Fine Line Laurie. Yes! You guys are right! It is the mathematical application for fractals in his honors geometry class. Sierpinski triangle within a delta symbol + variable x. Select Smaller Triangle #1. The family of generalised Sierpinski triangles is a set of four triangle shaped attractors found by generalising the iterated function system (IFS) of the Sierpinski triangle. This process is repeated over and over again with the resulting triangles to produce the Sierpinski triangle, as. How to build your triangle after installing this app: 1. Sierpinski. It is a self-similar structure that repeats at different levels of magnifications. However, we use a different method — Pascal’s triangle — to draw an approximation in Google Sheets. Eye of Magic. Then I recently wanted. Tags Level 7 Sierpinski Fractal Pyramid (Vase Mode) add to list. Upon each iteration, take each triangle which is pointing up and inscribe an. The Sierpinski triangle illustrates a three-way recursive algorithm. You may do so in. The Sierpinski triangle of order 4 should look like this: Related tasks. add to list. File history. Sierpinski by Kathryn Chan - The Sierpinski triangle is a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Sierpinski Fractal. Discover (and save!) your own Pins on PinterestThe Sierpiński triangle (sometimes spelled Sierpinski), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into. There are many designs and structures that use the fractal Sierpinski triangle to create materials with new optical properties [7,8], new magnetic properties [9], for temperature control [10–12], to generate molecular constructs [13,14] or to develop multiband antennas [15,16], which is an indicator of its potential applications in the industry. First we will begin with the process of repeated removal, and an exploration of the Sierpinski Triangle. Setup and calculation of the area of Sierpinski's Triangle4. 69 Sale. Example. Divide this large triangle into four new triangles by connecting the midpoint of each side. Sierpinski Triangle. a Sierpinski step on each triangle whose top node is labelled with the current generation number: the triangle is replaced by four triangles suc h that the top nodes of the three outer triangles. If this is done, the first few steps will look like this: If this is done an infinite number of times, its area. Soften the butter if you haven't already.