Evaluation of the Fractal Dimension of a Crystal
Abstract
The purpose of this experiment was to determine the effects of voltage and molarity changes on the fractal dimension of a Cu crystal formed by the re-dox reaction between Cu and CuSO4. Using the introductory information obtained from research, the fractal geometry of the Cu crystals was determined for each set of parameters. Through the analysis of data, it was determined that the fractal dimension is directly related to the voltage. The data also shows that the molarity is inversely related to the fractal dimension, but through research this was determined to be an error.
Introduction
A fractal is a geometric pattern that is repeated indefinitely that it cannot be represented with typical mathematics. Fractals can be seen in nature in the way minerals develop over time, the manner in which trees limbs shoot from the trunk, and the development of the human body (i.e. the lungs)1. These fractals determine a way to attempt to simplify the randomness of the universe via probability and theories regarding diffusion and intermolecular attractions.
The way dimensions in typical geometry are the typical 0-D, 1-D, 2-D, and 3-D. However, much matter does not fit these basic categories. A great example is a snowflake. If the negligible depth of a snowflake were ignored, it would be considered a 2-D object. However this is not completely true. A 2-D object can always be described by a finite number of tiles all in the same plane, because the snowflake cannot be described with only planes and also requires lines, it can be assumed it possesses properties of both a 1-D and 2-D object. A snowflake can be loosely approximated as a ~ 1.5-D object. This is fractal dimension of the object.
In order to determine a more exact fractal dimension of an object, smaller and smaller pieces are zoomed in upon and used to determine a rough estimate of the amount of pieces that exhibit the same pattern (self-similarity) as the whole object. The relationship between the zoom and self similarity of the object determine the fractal dimension:
It was proposed that if the length of the PVC pipes were to increase, then the sound produced will have a lower amplitude each time because the sound will lose energy as it continues in the pipe for a certain amount of time. However, the data actually showed that with every increase in pipe length, the amplitude got louder as well, thus refuting the hypothesis. These results made sense because what was created inside the PVC pipes was a standing still sound wave, or a resonance wave. These kinds of waves have certain locations on its wavelength in order for the change in sound to be heard, which it usually half a wavelength. With this, the tuning fork is 83.3Hz and a usual wavelength is about 300Hz, 300/83.3 = 3.6 meters, which is about 4 meters (half = 2 meters). So for the change in sound to be heard, the pipes had to be about 2 meters in change according to the frequency of the tuning
Sound is produced in a saxophone by the player providing a constant flow of air at a pressure above the atmosphere. The constant flow of air from the player is the source of energy, causing the air to oscillate creating vibrations in the air. The vibrating is created by the reed, which controls air flow through the mouthpiece. Also the vibrating is controlled by the resonance in the saxophone. The vibrating then radiates energy out of the open holes of the saxophone. The ...
Sound is a wave, which can change in pitch according to changing air pressure. It is produced by the vibrations of objects. Waves can be measured by speed (v), frequency (f), wavelength (λ), and period. The frequency of a wave describes how many cycles of a wave occur per unit time. A sound with a high frequency has more wavelengths in a set amount of time than a sound with a low frequency. High frequencies have smaller wavelengths, and low frequencies have larger wavelengths. The higher frequency, the higher the perceived pitch. The wavelength, frequency, and speed are all related by the equation: v = fλ. They can also be used in the equation: f = v/λ.
In contrast, there’s people who have pondered the question why do kids cheat in school? Some students blame family, coaches and teachers for pressuring them to do better and maintain a C plus average (Maecovitz 70-71). Some kids cheat because they want good grades, but are too lazy or busy with work to take the time to study with the teacher or to learn the materials (D’Aray 1). Other students cheat because they don’t kn...
Ultrasound is sound waves that have a frequency above human audible. (Ultrasound Physics and Instrument 111). With a shorter wavelength than audible sound, these waves can be directed into a narrow beam that is used in imaging soft tissues. As with audible sound waves, ultrasound waves must have a medium in which to travel and are subject to interference. In addition, much like light rays, they can be reflected, refracted, and focused.
momentum transfer when air molecules collide. Our ‘subjective impression’ about the frequency of a sound is called pitch. High pitch has high vibration frequency, while low pitch has a low vibration frequency. A pure musical tone consists of a single pitch or frequency. However, most musical tones are “complex summations” of various pure frequencies - one characteristic frequency, called the fundamental, and a series of overtones or harmonics Younger people can usually hear pitches with frequencies from about 20 hertz (infrasonic) to 20,000 (ultrasonic) hertz. We can’t hear above 20,000 hertz or below 20 hertz (ultra and infrasonic waves).
The vibration of the strings of a guitar causes the sound wave, but is not actually what you are hearing. The amplification of the sound wave is what is actually heard. The differences in the tension of the stings and the mass of the strings affect the pitch of the sound produced. The ends of each string are nodes, or where the wave does not travel from its initial position. The note you hear from the string is actually the first harmonic of the wave; other harmonics created when plucking a string form the undertones and overtones of a note. The waves on a guitar string are transverse waves, meaning they travel perpendicular to the original position. The waves are also standing waves, because they remain in the same position.
When viewing Georges Seurat’s, A Sunday on La Grande Jatte (Fig. 31-37), perception is changed vastly depending on the viewer 's proximity to the piece. At close range, all that is visible is a mass array of countless circular dots and tiny lines in a vast range of colors. Greens, blues, reds, oranges, yellows, white, browns, black and purples are all visible in a multitude of intensities. The Divisionism technique utilized causes this piece to appear as an abstract collection of colors when viewed at close range. Yet when distance is between the piece and the viewer, these seemingly sporadic dots come together to create a complete and detailed scene. Primarily consisting of biomorphic shapes, Seurat’s incorporates in every inch of the canvas
There are a few parts of sounds waves that we should be familiar with to better be able to understand the physics of music. The crest is the highest point of a wave, while the trough is the lowest. The wavelength of a wave is the distance between two adjacent ...
...Almost every student nowadays can admit to cheating at some point in their educational career, but motives as to why they cheat vary and can’t be narrowed down to one reason. There are many excuses as to why all levels of students chose to cheat no matter how wrong or unexpected it is. Cheating is a large problem, but members of universities and schools can discourage it and try to prevent it as much as possible. Not only should it be frowned upon, but it also should be a priority to make it next to impossible to do. Advisers can reduce the temptation of cheating by significantly decreasing the amount of true and false and multiple choice questions on exams and quizzes. Also by understanding the students social and parental pressures in a person at that age’s life is important and contributes to decreasing the amount of students cheating in schools and universities.
Each of the senses receives a different stimulus that allows us to perceive that specific type of information. For hearing the stimulus is sound waves. These are waves of pressure that are conducted through a medium (Martini, 2009). Often this medium is air but it can also be water or a solid object. Each wave consists of a region where the air molecules are gathered together and an opposite region where they are farther apart (Martini, 2009). A wavelength is the distance between either two wave peaks or two wave troughs. The number of waves that pass through a fixed reference point in a given time is the frequency. High pitch sounds have a high frequency where as low pitch sounds have a low frequency (Myers, 2010). The amplitude is the amount of energy, or intensity, in a sound wave. The more energy that a sound wave has, the louder it seems. For us to perceive any of the sound waves around us, they must pass through the external, middle, and inner ea...
The recursive sequence of numbers that bear his name is a discovery for which Fibonacci is popularly known. While it brought him little recognition during the course of his life, is was nearly 100 years later when the majority of the mathematicians recognized and appreciated his invention. This fascinating and unique study of recursive numbers possess all sorts of intriguing properties that can be discovered by applying different mathematical procedures to a set of numbers in a given sequence. The recursive / Fibonacci numbers are present in everyday life and they are manifested in the everyday life in which we live. The formed patterns perplex and astonish the minds in real world perspectives. The recursive sequences are beautiful to study and much of their beauty falls in nature. They highlight the mathematical complexity and the incredible order of the world that we live in and this gives a clear view of the algorithm that God used to create some of these organisms and plants. Such patterns seem not have been evolved by accident but rather, they seem to have evolved by the work of God who created both heaven and
Music is not purely a form of art. There is a great deal of science that goes behind the production of beautiful musical sounds. In order to understand how music is possible, one must have an understanding of physics. Physics allows us to create musical instruments with different tone qualities and the ability to be played in a certain way to produce a specific pitch or note. Music is sound, and sound is vibrations or waves that are at the right frequency to be perceived by the human ear. Audible vibrations are waves with a frequency between 16 and 20,000 vibrations per second. So what causes sound waves? Sound waves are caused by a disturbance in an elastic medium. These can include the strings on a violin, the reed on a clarinet, and even the human vocal cords. Click on the links below to discover how science makes it possible to create a variety of musical sounds.
What distinguishes sound waves from most other waves is that humans easily can perceive the frequency and amplitude of the wave. The frequency governs the pitch of the note produced, while the amplitude relates to the sound le...
Speaking of how the human ear receives music, sound is produced by vibrations that transmits energy into sound waves, a form of energy in which human ears can respond to and hear. Specifically, there are two different types of sound waves. The more common of the two are the transversal waves, which ...