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Sierpinski triangle number of triangles
sierpinski triangle number of triangles Sierpinski triangle. Does the "three-ness" of a triangle and the "four-ness" of a square seem to play a role in these numbers? Sierpinski triangle can be constructed through a number of different mathematical methods, however the most enjoyable is to draw it with pen and paper: 1. The formula for dimension $d$ is $n = m^d$ where $n$ is the number of self similar pieces and $m$ is the magnification factor. It should be noted now that in many of the At this level, we have 9 smaller black triangles remaining. The second iteration has 9 white triangles and 4 black triangles. A Sierpinski triangle can be created by starting with an equilateral triangle, breaking the triangle into 4 congruent equilateral triangles, and then removing the middle triangle. Nov 02, 2014 · The Sierpinski triangle is a fractal that consists of triangles, inside of triangles, inside of triangles and so on. We ﬁrst prove that the standard discrete Sierpinski triangle cannot strictly self-assemble in the Tile Assembly Model. To get the next step, connect the midpoints of the three sides to obtain four smaller triangles, three in the same orientation as the original and one upside down. Generalised Sierpinski triangles are interesting for a similar reason because they o er an extension to the classical Sierpinski triangle with fewer symmetries. Also note that the order of a Sierpinski triangle can be determined by counting the interior white triangles down the middle. Oct 18, 2012 · When describing how the area decreases you first state "splits each triangle into 4 congruent parts and by shading the central one, removes 1/4 of the area. a Sierpinski triangle by drawing in progressively smaller triangles. If your backing paper is large enough you can use 27 triangles to Aug 14, 2021 · Sierpinski triangle/Graphical . How many little triangles do we have after applying The Rule 2 times? 3 times? 20 times? times? We keep chopping little pieces out of . Jun 28, 2012 · The Sierpinski's triangle has an infinite number of edges. Your figure should appear as at right. All the images of Sierpinski's triangle have a finite number of iterations while in actuality the triangle has an infinite number of iteration. That’s the Sierpinski Triangle. The activity begins by considering observed patterns in number sequences and progresses to the concept of fractals, which is introduced to students through playing 'the chaos game'. Nov 13, 2013 · The Sierpinski Triangle The number of triangles after n iterations is 3n. Then, by connecting these midpoints smaller triangles have been created. So, in this case it could be called . “self-similar” which is a term we will visit later. May 27, 2021 · Sierpinski Triangle and Pascal’s Triangle. Do you see a pattern here? Use this pattern to predict what fraction of the triangle would be the total number of triangles remaining in the fourth iteration. For each red triangle at stage $n$ one additional white triangle appears at stage $n+1$. If the number of groups in your class does not allow for the creation of one Sierpinski Triangle, consider creating several smaller Sierpinski Triangles or combining triangles from groups in other . We start at stage 0, with an equilateral triangle with side length 1. Introduce the Sierpinski triangle. 6. Draw the points v1 to v∞. Look at how the pattern develops and ask pupils to create an instruction for what to do at each stage (eg. An order-n Sierpinski triangle, where n > 0, consists of three Sierpinski triangles of order n – 1, whose side lengths are half the size of the original side lengths, arranged so that they meet corner-to-corner. Jan 01, 2003 · Starting with "primitive" Pythagorean triangles, the text examines triangles with sides less than 100, triangles with two sides that are successive numbers, divisibility of one of the sides by 3 or by 5, the values of the sides of triangles, triangles with the same arm or the same hypotenuse, triangles with the same perimeter, and triangles . The key takeaway is that a seemingly complex pattern emerges from an extremely simple —… May 08, 2021 · Then Γ (n, 0) is an equilateral triangle with side 1, Γ (n, k) (k ≥ 1) is a union of smaller equilateral triangles with side 1 / n k and converges to K(n) as k → ∞ in the Hausdorff metric. The first 4 iterations of this process are shown below. Iteration 2: the more challenging problem of the strict self-assembly of discrete Sierpinski triangles, i. 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 Sierpinski triangle illustrates a three-way recursive algorithm. • Connect the midpoints to form an inverted (upside-down) triangle in the middle. If the first point v1 to lie . Divide that triangle into four equilateral triangles and . Recursively call the sierpinski method with each of the outside triangles, and the number of levels minus 1. The number of filled triangles at the nth iteration step is equal to 3ⁿ⁻¹, and their size is s×4¹⁻ⁿ, where s is the size of the initial triangle. Your poster now shows a Stage 4 Sierpinski Triangle. Apr 02, 2012 · SeirpTri(g, x1, y1, xa, ya, xb, yb, n - 1, count++); // recursively call the function using the 3 triangles SeirpTri(g, xa, ya, x2, y2, xc, yc, n - 1, count++); SeirpTri(g, xb, yb, xc, yc, x3, y3, n - 1, count++); return count; Sierpinski Triangle (Sierpinski Gasket) Grades: 5-8 Material: paper, ruler, protractor,crayons The Sierpinski Triangle, created in 1916 by Waclaw Sierpinski has some very interesting properties. What fraction of the triangle is the total number of triangles remaining in the third iteration? 27 / 64 8. Cut out of one tissue paper a triangle that is about one-half inch larger than the large triangle you cut out first. To find the perimeter of the whole triangle you have to take the sum of all the layers that are in the triangle, so it is the sum from k=1 to k= n of (3**(k-1))/(2**k). This is exactly the area of the inverted L-shape. Sierpinski Curve. It can grow or shrink by using the same pattern. It consists of an equilateral triangle, with smaller equilateral triangles recursively removed from its remaining area. 15 We call Γ (n, k) (k ≥ 0) the k-th approximating graph of general Sierpinski triangle K(n), in which vertex set and edge set are considered in a . Shrink the triangle by 1 2, make two copies, and position the three shrunk triangles so that each triangle touches each of the two other triangles at a corner. Repeat step 2 for the remaining triangles. Draw a triangle. This resource, from the Royal Institution, provides students with the opportunity to explore patterns in mathematics. Divide this large triangle into four new triangles by connecting the midpoint of each side. Let’s draw the first three iterations of the Sierpinski’s Triangle! Iteration 1: Draw an equilateral triangle with side . an equilateral triangle into four equilateral triangles removing the middle triangle and recursing leads to the Sierpinski triangle In three dimensions the Sierpinski triangle the Sierpinski carpet and the Sierpinski curve as are Sierpinski numbers and the associated Sierpinski problem. For this week's homework you will be working with this Geogebra Applet. 1. On a separate sheet, create Stage 3 of the . For context, the average width of the human hair is around 100,000 nanometres. In the first stage each side has been halved, four triangles are created and the centre one removed. // sierpinski triangle. It is an impressive and valuable topic for mathematical exploration. The Sierpinski Triangle may be formed in a manner analogous to the Cantor set. Take any equilateral triangle . It combines triangles and measurement The Sierpinski triangle generates the same pattern as mod 2 of Pascal's triangle. Wacław Sierpiński described the Sierpinski Triangle in 1915. The Sierpinski triangle, 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. Run several stages of the Sierpinski's Triangle and answer the following questions: Write down for each Stage: Number of Shaded Triangles. This is a nasty and forign looking . , the task of tiling a discrete Sierpinski triangle and nothing else. # sierpinski triangle. e. - Waclaw Sierpinski a Polish mathematician described the triangle in 1915 - he did not invent it - similar patterns are found in 13th-century mosaics and carpets - is starts with an equilateral triangle; View the Animated Construction Sierpinski triangle or demonstrate the process on a large chart paper. Level 4 has (27 * 3) = (9 * 9) = 81 tiny triangles. ” To build it “down,” start with a solid triangle and then remove the middle quarter, remove the middle . Explain that Aug 14, 2021 · Sierpinski triangle/Graphical . Nov 19, 2015 · 2015 is the 100th anniversary of the Sierpinski triangle, first described by Wacław Sierpiński, a Polish mathematician who published 724 papers and 50 books during his lifetime! The famous triangle is easily constructed by following these steps: Start with an equilateral triangle. A few years ago, we constructed the 8th order version of this fractal, comprised of 3^8 individual triangles (6,561). Area of one Shaded Triangle. 5849625 Sierpinski Rectangle an equilateral triangle into four equilateral triangles removing the middle triangle and recursing leads to the Sierpinski triangle In three dimensions the Sierpinski triangle the Sierpinski carpet and the Sierpinski curve as are Sierpinski numbers and the associated Sierpinski problem. An algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle is as follows: 1. Two iteration of Sierpinski triangle into four smaller triangles and again removing the middle triangle from each of the larger triangles. Above are the first five stages of the Sierpinski triangle. Start with an equilateral triangle with a base parallel to the horizontal axis. The third iteration has 27 white triangles and 13 black triangles. Hello Class. The first and last segments are either parallel to the original segment or meet it at 60 degree angles. The number of smallest triangles triples again in the third iteration, to 27. When we look at the finished Sierpinski Triangle, we can zoom in on any of these three sub-triangles, and it will look exactly like the entire Sierpinski Triangle itself. This design is called Sierpinski's Triangle (or gasket), after the Polish mathematician Waclaw Sierpinski who described some of its interesting properties in 1916. It should be noted now that in many of the Another Way to Create a Sierpinski Triangle- Sierpinski Arrowhead Curve. 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. Also, the number of all triangles in N-W triangular part of the canvas is always the same. The output below shows the correct number of triangles based on the number of levels. , which is named after the Polish mathematician Wacław Sierpiński. Sierpinski triangle can be constructed through a number of different mathematical methods, however the most enjoyable is to draw it with pen and paper: 1. Number of Shaded Triangles 7. 1 . These carbon molecule structures created a sort of "muffin tin" that trapped the electrons in the Sierpinski triangle structure. Wacław Franciszek Sierpiński (1882 – 1969) was a Polish mathematician. In fact, Pascal's triangle mod 2 can be viewed as a self similar structure of triangles within triangles, within triangles, etc. Starting from a single black equilateral triangle with an area of 256 square inches, here are the first four steps: . A Sierpinski triangle shows a well-known fractal structure. I. The key takeaway is that a seemingly complex pattern emerges from an extremely simple —… If the number of groups in your class does not allow for the creation of one Sierpinski Triangle, consider creating several smaller Sierpinski Triangles or combining triangles from groups in other . This is the only triangle in this direction, all the others will be upside down: Inside this triangle, draw a smaller upside down triangle. Oct 31, 2016 · The Sierpinski triangle, like many fractals, can be built either “up” or “down. But even if the number of triangles increases dramatically the rules don´t change. Continue in this way. Therefore, Sierpinski gasket's area is 0. To construct a Sierpinski Triangle, first draw an equilateral triangle. Iteration 2: Jan 25, 2012 · When shifting to 2 dimensions, starting with a triangle, dividing it up into 4 similar smaller triangles and removing the middle triangle results in the Sierpinski Gasket; the limit of colouring Pascal's triangle with the even numbers as black and the odd numbers as white, as the number of rows tends to infinity is the Sierpinski Gasket. Example. Total Shaded Area. Use the following iterations (or steps) to create a famous fractal based on the equilateral triangle called the Sierpinski triangle. So, the perimeter of the newest set of triangles is (3**(n-1))/(2** n). Choose a tissue paper color for each size triangle that you have cut out. surrounding an empty space created by an equilateral triangle of the same size. What patterns do you see in the numbers for the number of shaded triangles? Jan 25, 2012 · When shifting to 2 dimensions, starting with a triangle, dividing it up into 4 similar smaller triangles and removing the middle triangle results in the Sierpinski Gasket; the limit of colouring Pascal's triangle with the even numbers as black and the odd numbers as white, as the number of rows tends to infinity is the Sierpinski Gasket. If your backing paper is large enough you can use 27 triangles to The Sierpinski number program is frequently asked in Java coding interviews and academics. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral of Anagni, Italy. Repeat this process with the nine remaining triangles. Magdalena. The pictures of Sierpinski's triangle appear to contradict this; however, this is a flaw in finite iteration construction process. The pedal triangle divides the original triangle into four smaller triangles. An example is shown in Figure 3. Lab 7: Sierpinski Fractals and Recursion. Its dimension is fractional—more than a line segment, but less than a . If the test is successful, the inner loop expand! performs a Sierpinski step on each triangle whose top node is labelled with the current generation number: the triangle is replaced by four triangles such that the top nodes of the three outer triangles are labelled with the next higher generation Generation of Sierpinski Triangles 533 number. The number of triangles removed at each step is multiplied by 3 whereas the area of any one triangle removed is divided by 4. For example, here are Sierpinski triangles of the first few orders: The first step in the geometric construction of the Sierpinski Triangle involved splitting a triangle up into three other triangles. The sides of each triangle are one half the length of the triangles in the previous iteration, so the formula for the perimeter is P 1 2 n, where P is the perimeter of the original . That is to say, the even numbers in Pascal's triangle correspond with the white space in Sierpinski's triangle. Oct 06, 2015 · This means that the the number of triangles at each layer is 3**(n-1). Just cut the middle triangle out. It uses three different colors to draw it - white for triangles' border, brown for background and red for inner triangles. Jan 25, 2012 · When shifting to 2 dimensions, starting with a triangle, dividing it up into 4 similar smaller triangles and removing the middle triangle results in the Sierpinski Gasket; the limit of colouring Pascal's triangle with the even numbers as black and the odd numbers as white, as the number of rows tends to infinity is the Sierpinski Gasket. The above program will allow the user to create their own Sierpinski Triangle, and watch the fractal develop as they step through each recursive iteration. The Sierpinski Curve is a base-motif fractal that looks exactly like the Sierpinski Triangle after an infinite number of iterations. from each vertex drop a perpendicular to the opposite side until it intersects with that side. Subdivide into four smaller triangles (split the edges of the first triangle) 3. If the first point v1 was a point on the Sierpiński triangle, then all the points vn lie on the Sierpinski triangle. Among these is its fractal or self-similar character. The Sierpiński Triangle: The Sierpiński Triangle is an example of a fractal pattern, since smaller versions of the same shape occur within the triangle. Start by labeling p1, p2 and p3 as the corners of the Sierpinski triangle, and a random point v1. The Sierpinski triangle of order 4 should look like this: Related tasks. Start with a single large triangle. The concept of the Sierpinski triangle is very simple: Take a triangle; Create four triangles out of that one by connecting the centers of each side; Cut out the middle triangle; Repeat the process with the remaining triangles; Mathematical aspects: The area of the Sierpinski Triangle approaches 0. Add the outside triangles in the following order: top, left, and right. This procedure is continued indefinitely. The large blue triangle consists of three smaller blue triangles, each of which itself consists of three smaller blue triangles . Sierpinski triangle or gasket triangle is a gematrical object that . Produce an ASCII representation of a Sierpinski triangle of order N. Use the Sierpinski 1 macro to create a second iteration Sierpinski Triangle by clicking on each of the lines joining the midpoints. Start with a colored triangle, a Stage 0 Sierpinski triangle. When you run the main method in Recursion, it will display all triangles. Start with one line segment, then replace it by three segments which meet at 120 degree angles. The Sierpinski triangle is a self-similar fractal. Ignore the central triangle (s). This case study is mostly a performance benchmark, involving the construction of all triangles up to a certain number of iterations . 3. The Sierpinski triangle is a fractal; if any of the smaller right-side-up triangles within the Sierpinski triangle were to be blown up to the size of the Sierpinski triangle itself, the same image would be viewed. Sierpinski Triangle Replacing N by three ( as each iteration creates three self-similar triangles) and r by two ( as the sides of the triangles are divided by two) in the Hausdorff-Besicovitch equation gives: D = log(3) / log(2) = 1. 2 . A Sierpinski triangle has interesting topological and dimensional properties, which can be 4. This is because with every iteration 1/4 of the area is taken away. Set vn+1 = 1 / 2 (vn + prn), where rn is a random number 1, 2 or 3. n n SS . Sierpinski enrolled Sierpinski curves are a recursively defined sequence of continuous closed plane fractal . We can also apply this definition directly to the (set of white points in) Sierpinski triangle. Help us Create the World’s Largest Fractal Sierpinski Triangle! Every year we build a giant fractal triangle made of thousands of individual fractals triangles made by students all over the world. Number sequences can be observed in each of the "Fractal Doodles" below as well. We can break up the Sierpinski triangle into 3 self similar pieces $(n=3)$ then each can be magnified by a factor $m=2$ to give the entire triangle. It may not be obvious from these illustrations that inside each larger triangle, three (not one) smaller triangles are drawn. Sierpinski's Triangle Exploration Questions Run several stages of the Sierpinski's Triangle Activity, and answer the . Select each of the final little triangles as the final objects for a new macro. • Color in this triangle. Instructions: A) Run several stages of the Sierpinski's Triangle B) Answer the following questions in your notebook: 1) Write down for each Stage the number of Shaded Triangles 2) Pattern 1: 1, 3, 9, 27 a) Explain what this sequence represents in the Sierpinski’s triangle? May 10, 2015 · The Sierpinski Triangle is a fractal, with the overall shape of an equilateral triangle. The first step in the geometric construction of the Sierpinski Triangle involved splitting a triangle up into three other triangles. Use the Sierpinski 1 macro to create a first iteration Sierpinski Triangle. Sierpinski triangle/Graphical for graphics images of this pattern. For instance, in the diagram labeled "2 iterations," one smaller triangle has been drawn in each corner of the larger triangle; the smaller triangle that appears in the middle is . Sierpinski sieve generator examples Click to use. Feb 15, 2016 · The Sierpinski triangle is a fractal with the form of a triangle subdivided recursively into smaller ones. On the last page is an image of isometric dot paper which is particularly useful for creating equilateral triangles - and an image of the Initiator has been drawn for you. We start with the entire (solid) equilateral triangle. 3 . To get to know the Sierpinski Triangle better, here’s some food for thought: When we applied The Rule for the first time, we started with one triangle and ended up with three. Iteration 1: • Using a centimeter ruler, find the midpoint of each side of the triangle. Nov 24, 2019 · Sierpinski’s triangle is an algorithm that demonstrates an interesting property of randomness (Python). It is a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. 5. Oct 28, 2010 · The base in the Sierpinski triangle has 1 white triangle and zero black triangles. Before moving ahead in this section, first, we will understand the Sierpinski triangle because the Sierpinski number is closely connected to the Sierpinski triangle. The outside of this removed triangle still remains and is part of the perimeter of the Sierpinski triangle. It combines triangles and measurement The number of red triangles at each stage is multiplied by three to give the number of red triangles at the next stage. Sierpinski Triangle¶ Another fractal that exhibits the property of self-similarity is the Sierpinski triangle. Sierpinski Triangle. Can we obtain the same result when removing triangular tremas? Assume the whole triangle's area is 1. When done you should have a figure that looks like iteration 3 of Sierpinski’s triangle. ‘find the largest upright triangle and draw in the largest downward-pointing equilateral triangle’). Create a 4th Order Sierpinsky Triangle. Here is a basic image of what it looks like… Usually, when you think of this shape, you think of it as being built by small triangles in such a way that former larger and larger ones. The pedal triangle of an acute triangle T is the triangle formed by the three points that lie at the feet of the three altitudes of T, i. 8. An order-0 Sierpinski triangle is a plain filled triangle. At each stage, we divide each solid triangular portion into four similar triangles and remove the central one. To create a Stage 1 triangle, connect the midpoints of the sides to form four smaller triangles; color the three outer triangles and make the inner one white. (The rst time this is asked is after 2 iterations, for a total of 9 unshaded triangles). Nov 13, 2018 · The resulting Sierpinski triangles are jaw-droppingly small - less than 20 nanometres across one side. A Sierpinski triangle is, “A pattern has begun by finding the midpoints of the line segments of the largest triangle. Fractals are self-similar patterns that repeat at different scales. Required options. The total area of the red triangles at each stage is multiplied by 3/4 to give the total area of the red triangles at the next stage. The Sirpenski triangle is composed of multiple triangles inside of one triangle. To make a Sierpinski triangle, take an equilateral triangle, then draw three small triangles, one in each angle, then repeat it with those 3 new triangles, and repeat it again, and again. (3 / 4)2 = 9 / 16 (3 / 4)3 = 27 / 64 (3 / 4)4 = 81 / 256 9. Feb 05, 2015 · So, the triangle counts of Sierpinski triangles are 1, 3, 9, 27, 81, 243 and 729. Jan 07, 2021 · I discovered this algorithmic approach for creating the Sierpinski triangle fractal (ironically) by doing the math and found that the centres of each of the three new 1/2 scaled triangles connected together formed another 1/2 scale triangle. It is a self-similar structure that repeats at different levels of magnifications. The Sierpinski triangle generates the same pattern as mod 2 of Pascal's triangle. This pattern is then repeated for the smaller triangles, and essentially has infinitely many possible iterations. Do this again to each of the shaded triangles in the Stage 1 triangle to get the Stage 2 triangle. Sierpinski Triangle in Galilean Plane 153 (a) S 0 (b) S 1 (c) S 2 Figure 1. Now there are four new triangles. For convenience, draw it with one side parallel to the bottom of your paper, so the top is a vertex. Ignoring the middle triangle that you just created, apply the same procedure to . What do you think happens to these numbers as the number of stages approaches infinity? Compare these results to those for the Sierpinski's Carpet . The procedure for drawing a Sierpinski triangle by hand is simple. volume 4 (2011), number 4 115 triangles, and so on, obtaining, at level n, a set Sn consisting of 3 n equilateral triangles. . 4. Each student will make their own fractal triangle composed of smaller and smaller triangles. Create Stage 2 of the Sierpinski triangle. The Sierpinski Triangle S, or Sierpinski Gasket, is the limit set of this procedure, i. Sierpinski Triangle (Sierpinski Gasket) Grades: 5-8 Material: paper, ruler, protractor,crayons The Sierpinski Triangle, created in 1916 by Waclaw Sierpinski has some very interesting properties. Number of Shaded Triangles Jan 07, 2021 · I discovered this algorithmic approach for creating the Sierpinski triangle fractal (ironically) by doing the math and found that the centres of each of the three new 1/2 scaled triangles connected together formed another 1/2 scale triangle. Repeat step 2 for each of the remaining smaller triangles forever. Use 3 more learner’s triangles to form another Stage 4 triangle on the bottom line touching the first one, then another three triangles on top to make a Stage 5 triangle, using altogether 9 triangles like the orange diagram. The Sierpinski triangle is a fractal described by Waclaw Sierpinski in 1915. 2. the Sierpinski triangle: The construction of the first level Sierpinski triangle is easy: Take a regular triangle and connect the middle-points of the sides. Thus, the dimension of a Sierpinski triangle is log (3) / log (2) ≈ 1. The first iteration has 3 white triangles and 1 black triangle. Three white triangles down the middle indicates an order 3 Sierpinski triangle: Next, the Koch pattern can be analyzed. The limiting shape has zero area but the sum of all triangle edges has infinite length. This example iterates Sierpinsky algorithm for 4 iterations and draws it on a 400- by 400-pixel canvas. 585. Following this pattern, how many white triangles will the 15th iteration have? May 27, 2021 · Sierpinski Triangle will be constructed from an equilateral triangle by repeated removal of triangular subsets. ”. " These two shaded areas are are the triangle and the negative space. This process is then repeated for each of the triangles, and each of those triangles, and so on, creating the shape shown above. Divide it into 4 smaller congruent triangle and remove the central triangle . This wikipedia page talks about it in some detail and shows several different ways of building the triangle. Sierpinski Triangle The Sierpinski Triangle, also called Sierpinski Gasket and Sierpinski Sieve, can be drawn by hand as follows: Start with a single triangle. Task. An IFS and an Sierpinski Pedal Triangle. The progression continues tripling, and the number of triangles, T, can be expressed as: T=3 n Sierpinski's Triangle Exploration Questions Run several stages of the Sierpinski's Triangle Activity, and answer the . The Sierpinski Triangle & Functions The Sierpinski triangle is a fractal named after the Polish mathematician Waclaw Sierpiński who described it in 1915. Sierpinski Triangles Investigation. We can decompose the unit Sierpinski triangle into 3 Sierpinski triangles, each of side length 1/2. " Later you state "The Sierpinski's triangle has total area of 0 (defining area as the shaded region). Upon calling the sierpinski command at the AutoCAD command-line, the program will prompt the user to specify three distinct non-collinear points defining an arbitrary . sierpinski triangle number of triangles