The Geometry of Nature, Real World Entities, and Fractals

Concepts in Mathematics

Chapter 3) The Geometry of Nature, Real‑World Entities, and Fractals

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After I complete a writing task, I select a number of websites from other authors, and link to them, to provide additional information and alternative perspectives for the reader. The links are the blue underlined words, and they can be seen throughout this book. The in‑line links, such as the link on these words, are primarily to support the material I wrote, or to provide additional details. The links presented at the end of some of the paragraphs, subsections, and sections are primarily for websites with additional information, or alternative points of view, or to support the material I wrote. The websites contain articles, videos, and other useful material.

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The Geometry and Structure of Nature, and Real‑World Entities, with Related Concepts

The Geometry of Nature, and Real-World Entities

The geometry found in nature, is very different from the idealized geometry of circles, squares, isosceles triangles, spheres, pyramids, and cubes. The idealized geometry that mathematicians traditionally studied is relatively simple, very orderly, and often symmetrical. However, the geometric structures found in nature are usually highly complex, and may appear to be disorderly, or random. Some examples are the geometric structures of mountains, rivers, valleys, canyons, boulders, rocks, plants, and animals.

The geometry of two entities from the same species may contain some geometric similarities, coupled with a large number of differences. For example, two pine trees might have some similarities, such as in texture, leaf structure, and overall appearance, but they certainly do not have identical geometries. This obvious concept can be seen at Google images, by left clicking on this link: “Pine Trees.”

Nature’s Geometry, can be Understood by Studying Real-World Entities at Different Levels of Magnification

Nature’s geometry can be understood, by examining the structure of real-world entities at different levels of magnification, such as the cellular, molecular, and atomic, levels, as well as with the naked eye. When this is done, it becomes apparent that real-world objects, whether they are manufactured, or produced by nature, are comprised of tiny building blocks that are three dimensional. Some examples of building blocks are cells, tiny crystals, as well as molecules, and atoms. These building blocks are arranged in precise configurations, to form a specific geometric structure.

The building blocks that create structures are composed of even smaller building blocks. The smaller building blocks are generally VERY DIFFERENT from the larger building blocks they comprise. For example, sodium chloride is common table salt, which is comprised of small crystals, which are comprised of still smaller crystals. However, these crystals are ultimately comprised of sodium, and chlorine, which are joined together with strong chemical bonds. Sodium and chlorine do not resemble table salt in any way, and both are extremely corrosive and poisonous. Sodium is actually an extremely reactive metal, with a grayish color, and chlorine has a light green color. This certainly does not resemble table salt.

An important consideration with any type of structure is the strength and overall nature of the bonding forces that are holding the building blocks together. For example, hydrogen and oxygen when mixed together is extremely explosive. However, when hydrogen and oxygen are chemically bound together to form water, the molecular geometry changes, and the explosive properties are eliminated.

In general, when you examine any object at varying levels of magnification, you will observe an entirely different set of structures, comprised of a different set of building blocks. Even if the original structure had very disorderly geometry, as the magnification increases, you will notice increasing levels of orderly arrangements comprised of smaller and smaller building blocks. Generally, each level of magnification will reveal unique geometric structures that do not resemble the original entity, or the structures viewed at lower levels of magnification.

There is one important exception to the concept presented in the previous paragraph, which are crystals. Generally, a crystal is comprised of smaller, and smaller, and still smaller crystals. However, when you reach the molecular or atomic levels, the building blocks are no longer crystals. They are of course molecules and atoms.

The Ideas Presented in this Section Involving Nature’s Geometry, can be SEEN at the Following Websites:

The following searches are from Google images:

See Human skin and compare with Human skin Microscope

See Leaf and compare with Leaf Microscope

See carbon and compare with carbon Microscope, and also

“carbon atoms” electron scanning microscope

The following two websites are very interesting. Both of these sites have software devices comprised of slides, where you can control the magnification of the entities displayed down to the subatomic level. Most of the images appear to be ACCURATE artistic renditions. A few of the images appear to be real photographs. 1) Zooms in on a leaf, from meters, to meters, from the National High Magnetic Field Laboratory, Contributors Matthew J. Parry-Hill, and others, 2) Zooms in on the eye of a fly from meters, to meters, the Universe-Solved! Version of "The Powers of 10". NOTE: The images from the longest distances simulated on these websites are obviously not photographs, because it would be impossible to see the sun or even the entire solar system, from or even meters.

Fractals Applied to the Geometry of Nature, and Matter

An Introductory Look at Fractals

A fractal is a mathematical concept that involves complex geometric structures, which are usually generated by a computer, using special software, such as the following examples:

The above examples, and all the other fractals in this chapter are from a free computer program, called with XaoS. This program can be downloaded from the following URL: http://fractalfoundation.org/resources/fractal-software.

Note: all the fractals, and graphs in this chapter were enhanced with the functions in Microsoft Word. This involved cropping, and modifications in contrast, brightness, and/or color.

What Are Fractals?

Fractals are geometric figures that either, do not change in appearance with increasing magnification, except for color, or if there is a change at one level of magnification, the original geometric structure will eventually reappear at a higher level of magnification.

That is fractals are essentially comprised of increasingly smaller and smaller building blocks, which produce the same geometric patterns, over-and-over-again, as the magnification increases, and approaches infinity. However, as the magnification increases the colors of a fractal usually change. The specific color displayed is programmed into the software that is generating the fractal.

Some fractals can involve a series of structures resembling the branches on a tree, where one large branch, leads to a set of small branches, and each of the small branches leads to another set of still smaller branches. This sequence of smaller and smaller branches does not stop, which is the same as saying it continues to infinity.

For examples of fractals, see the websites presented below, and the following subsection: 1) Fractals from Google images

2) Branching fractals from Google images

An Example of a Fractal, at Different Levels of Magnification, Presented in 25 Screenshots

Below this paragraph there 25 images of a fractal as it increases in magnification. The first image (1) is not magnified, and the remaining 24 images are magnified sequentially. I estimate, that the last image was magnified several thousand times if not more. However, keep in mind that we are dealing with a mathematical concept, on a computer screen, and the magnification is essentially a simulation produced by the software.

The limits of the software I was using was apparent after the 25^th image, where the screen turned black. If this limitation was not present, all the patterns presented below would eventually reappear at higher levels of magnification.

At this point, the limits of the software were reached, and with additional magnification, the screen turned black.

For additional and/or supporting information, see the following websites from other authors: 1) Video: Deepest Mandelbrot Set Zoom Animation ever - a New Record! 10^275 (2.1E275 or 2^915), 2) Video: Best fractals zoom ever, 3) Mandelbrot Zoom 10^227 [1080x1920], 4) Mandelbrot fraktaler super deep 2 2^4750.

Fractals Are Mathematical Concepts, and they Should Not Be Confused with Real-World Entities

Many sources erroneously imply that fractals are structures found in nature. This is usually coupled with claims that mountains, trees, and the blood vessels in animals and humans are comprised of fractals. Physical structures comprised of fractals CANNOT exist in nature, and they CANNOT be created by humans. Fractals exist in the human mind, and on the computer screens. Fractals are mathematical conceptualizations that have properties that cannot exist in the real world. This includes infinite repeating patterns at different levels of magnification, and infinite perimeters for fractals that are not composed of perfect geometric figures. With Real-world entities, every level of magnification USUALLY reveals a different set of geometric structures, with only a few exceptions, such as crystals.

However, fractals can be used to APPROXIMATE the geometry, structures, and texture of many real‑world entities, including mountains, trees, roots, blood vessels, as well as entities created by humans. Fractals are especially useful in duplicating textures, and the highly complex geometries of nature, in graphic design, and animations. In general, fractals represent a new concept in geometry (fractal geometry) that has very wide theoretical and practical applications.

Representing the Structure and Geometry of Nature: with Fractals, versus Raster Graphics

As explained above, the complex geometric structures of nature can be represented by fractals, but this is actually a less-than-perfect approximation. Usually the approximation is relatively poor, when compared to raster graphics. Fractals, and raster graphics involve mathematical concepts, and the computer. However, raster graphics are produced by digital photographic equipment that converts images into mathematics that can only be processed by a computer. In general, when precise detail of complex geometric structures is needed, the ideal method is to use raster graphics. This is especially the case when dealing with the textured and irregular geometric structures of nature.

One of the most important differences between fractals and raster graphics involves the complexity of the mathematics. Fractals, can be produced with relatively simple equations that can be understood by anyone with a mathematical background. These equations are interesting from a mathematical perspective. The mathematics involved with raster graphics can only be processed by a computer, because a massive number of calculations are involved. Each pixel comprising an image of a raster graphic, involves calculations that relate to coordinates and color. A high-resolution raster graphic may contain OVER 8 million pixels. These calculations differ for each image that is produced, and they are somewhat repetitive, and not particularly interesting from a mathematical point of view. The mathematics of raster graphics cannot be converted into a few simple equations.

For example, the following fractal was produced with the computer with the simple formula shown above it. However, if the fractal were converted to a raster graphic, the mathematics would be too lengthy to comprehend.

Methods of Producing Fractals

There are many methods of producing fractals. One of the simplest involves line drawings of geometric structures carried out with the following online software: http://sciencevsmagic.net and http://sciencevsmagic.net/fractal. Another technique of creating fractals involves graphing equations with complex numbers, in repetitive sequence, using different sets of numbers. This does not resemble conventional graphing techniques, and it is carried out by a computer, using software designed for the purpose. This technique was devised by Benoit Mandelbrot, in the late 70s, while he worked at IBM. See also Fractals and Benoit Mandelbrot, and the Story of Benoit B. Mandelbrot and the Geometry of Chaos.

Eight Fractals Generated with Formulas

In this subsection, the technique devised by Benoit Mandelbrot is used to generate eight fractals. The technique is carried out by a computer using software called XaoS. Seven of the eight formulas used to generate the fractals I devised. XaoS displays formulas as shown in the screenshot at the end of this paragraph, which is: z^2+c. For simplicity, and convenience I am presenting the formulas I used, as equations such as the following: .

I show a conventional three-dimensional graph of each equation, I used to generate a fractal. This is to illustrate that there are NO similarities between conventional graphing methods, and the techniques used to generate fractals. The fractals are generated with complex numbers, and the conventional graphs do not involve complex numbers, or the repetitive graphing technique the computer uses to generate fractals. The conventional grafts were created with Microsoft mathematics add-in for Word.