{"id":1398,"date":"2023-04-06T14:49:12","date_gmt":"2023-04-06T14:49:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=1398"},"modified":"2025-05-22T18:48:38","modified_gmt":"2025-05-22T18:48:38","slug":"fractal-basics-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractal-basics-learn-it-2\/","title":{"raw":"Fractal Basics: Learn It 2","rendered":"Fractal Basics: Learn It 2"},"content":{"raw":"

Constructing a Fractal<\/h2>\r\n

We can construct other fractals using a similar approach of recursion. To formalize this a bit, we\u2019re going to introduce the idea of initiators <\/strong>and generators<\/strong>.<\/p>\r\n

\r\n
\r\n

initiators and generators<\/h3>\r\n

An initiator<\/strong> is a starting shape.<\/p>\r\n

 <\/p>\r\n

A generator<\/strong> is an arranged collection of scaled copies of the initiator.<\/p>\r\n<\/div>\r\n<\/section>\r\n

To generate fractals from initiators and generators, we follow a simple rule, the fractal generation rule.<\/p>\r\n

\r\n
\r\n

fractal generation rule<\/h3>\r\n

At each step, replace every copy of the initiator with a scaled copy of the generator, rotating as necessary.<\/p>\r\n<\/div>\r\n<\/section>\r\n

This process is easiest to understand through an example.<\/p>\r\n

Use the initiator and generator shown to create the iterated fractal.
\"A<\/center>\r\n

 <\/p>\r\n\r\nThis tells us to, at each step, replace each line segment with the spiked shape shown in the generator. Notice that the generator itself is made up of [latex]4[\/latex] copies of the initiator. In step [latex]1[\/latex], the single line segment in the initiator is replaced with the generator. For step [latex]2[\/latex], each of the four line segments of step [latex]1[\/latex] is replaced with a scaled copy of the generator:
\r\n

\"Step<\/center>\r\n

 <\/p>\r\n\r\nThis process is repeated to form Step [latex]3[\/latex]. Again, each line segment is replaced with a scaled copy of the generator.
\r\n

\"Step<\/center>\r\n

 <\/p>\r\n\r\nNotice that since Step [latex]0[\/latex] only had [latex]1[\/latex] line segment, Step [latex]1[\/latex] only required one copy of Step [latex]0[\/latex]. Since Step [latex]1[\/latex] had [latex]4[\/latex] line segments, Step [latex]2[\/latex] required [latex]4[\/latex] copies of the generator. Step [latex]2[\/latex] then had [latex]16[\/latex] line segments, so Step [latex]3[\/latex] required [latex]16[\/latex] copies of the generator. Step [latex]4[\/latex], then, would require [latex]16\\cdot4=64[\/latex] copies of the generator.
\r\n
\r\nThe shape resulting from iterating this process is called the Koch curve, named for Helge von Koch who first explored it in 1904.\"A<\/section>\r\n

Use the initiator and generator shown to produce the next two stages.
\"Initiator<\/center>[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"703380\"]
\"Step<\/center>[\/hidden-answer]<\/section>\r\n

Using iteration processes like those above can create a variety of beautiful images evocative of nature. More natural shapes can be created by adding randomness to the steps.<\/p>\r\n

Create a variation on the Sierpinski gasket by randomly skewing the corner points each time an iteration is made. Suppose we start with the triangle below. We begin, as before, by removing the middle triangle. We then add in some randomness.
\"Step<\/center>\r\n

 <\/p>\r\n\r\nWe then repeat this process.

\"Step<\/center>\r\n

 <\/p>\r\n\r\nContinuing this process can create mountain-like structures. This landscape[footnote]http:\/\/en.wikipedia.org\/wiki\/File:FractalLandscape.jpg[\/footnote]\u00a0was created using fractals, then colored and textured.

\"A<\/center><\/section>\r\n
[videopicker divId=\"tnh-video-picker\" title=\"Fractals\" label=\"Select Video\"]
\r\n[videooption displayName=\"What Is A Fractal (and what are they good for)?\" value=\"https:\/\/youtu.be\/WFtTdf3I6Ug\"][videooption displayName=\"Fractals are typically not self-similar\" value=\"https:\/\/youtu.be\/gB9n2gHsHN4\"] [videooption displayName=\"How fractals can help you understand the universe - BBC Ideas\" value=\"https:\/\/youtu.be\/w_MNQBWQ5DI\"]
\r\n[\/videopicker]\r\n\r\n

 <\/p>\r\n

You can view the transcript for \u201cWhat Is A Fractal (and what are they good for)?\u201d here (opens in new window).<\/a><\/p>\r\n

You can view the transcript for \u201cFractals are typically not self-similar\u201d here (opens in new window).<\/a><\/p>\r\n

You can view the transcript for \u201cHow fractals can help you understand the universe | BBC Ideas\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n

 <\/p>","rendered":"

Constructing a Fractal<\/h2>\n

We can construct other fractals using a similar approach of recursion. To formalize this a bit, we\u2019re going to introduce the idea of initiators <\/strong>and generators<\/strong>.<\/p>\n

\n
\n

initiators and generators<\/h3>\n

An initiator<\/strong> is a starting shape.<\/p>\n

 <\/p>\n

A generator<\/strong> is an arranged collection of scaled copies of the initiator.<\/p>\n<\/div>\n<\/section>\n

To generate fractals from initiators and generators, we follow a simple rule, the fractal generation rule.<\/p>\n

\n
\n

fractal generation rule<\/h3>\n

At each step, replace every copy of the initiator with a scaled copy of the generator, rotating as necessary.<\/p>\n<\/div>\n<\/section>\n

This process is easiest to understand through an example.<\/p>\n

Use the initiator and generator shown to create the iterated fractal.<\/p>\n
\"A<\/div>\n

 <\/p>\n

This tells us to, at each step, replace each line segment with the spiked shape shown in the generator. Notice that the generator itself is made up of [latex]4[\/latex] copies of the initiator. In step [latex]1[\/latex], the single line segment in the initiator is replaced with the generator. For step [latex]2[\/latex], each of the four line segments of step [latex]1[\/latex] is replaced with a scaled copy of the generator:<\/p>\n

\"Step<\/div>\n

 <\/p>\n

This process is repeated to form Step [latex]3[\/latex]. Again, each line segment is replaced with a scaled copy of the generator.<\/p>\n

\"Step<\/div>\n

 <\/p>\n

Notice that since Step [latex]0[\/latex] only had [latex]1[\/latex] line segment, Step [latex]1[\/latex] only required one copy of Step [latex]0[\/latex]. Since Step [latex]1[\/latex] had [latex]4[\/latex] line segments, Step [latex]2[\/latex] required [latex]4[\/latex] copies of the generator. Step [latex]2[\/latex] then had [latex]16[\/latex] line segments, so Step [latex]3[\/latex] required [latex]16[\/latex] copies of the generator. Step [latex]4[\/latex], then, would require [latex]16\\cdot4=64[\/latex] copies of the generator. <\/p>\n

The shape resulting from iterating this process is called the Koch curve, named for Helge von Koch who first explored it in 1904.\"A<\/section>\n

Use the initiator and generator shown to produce the next two stages.<\/p>\n
\"Initiator<\/div>\n