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
The Helmholtz resonance of a guitar is due to the air at the sound hole oscillating, driven by the springiness of the air inside the body. This is analysed quantitatively in Helmholtz Resonance.
Sound is created by vibrations that travel in waves, with the longer, more spread out waves being lower pitched sound, and shorter, closer together waves being higher pitched sound. Sound waves travel through the air or water (or whatever is in their way) to reach your ear and vibrate your eardrum, which in turn lets you hear.Sound can not be heard in space because there is no air for it to travel on. Volume is measured in decibels, with a pin drop at about 15 decibels, a 12-gauge shotgun being at about 160 decibels, and a normal conversation at about 65 decibels.
The Holocaust was a time in history that was brutal, sickening, and ruthless. The Holocaust refers to, “The systematic, bureaucratic, state-sponsored persecution and murder of approximately six million Jews by the Nazi regime and its collaborators.” (ushmm.org) The Nazi’s came to rule in January of 1933 in Germany. They supposed that the German’s were “ethnically greater” and the Jews considered “mediocre” were a threat and problem to the German society. The German’s did not only target the Jews, they targeted Gypsies, Slavic people, Russians, the disabled, homosexuals, Jehovah’s Witnesses, etc. If those people were captured but not killed yet, they were sent away to concentration camps. Concentration camps were introduced as harsh, uncleanness living camps and a very important feature of the regime. After Adolf Hitler became leader in January of 1933, he developed the first concentration camps in Germany (ushmm.org).
Everything in the universe involves some type of physics. Even the universe itself does, but have you ever wondered about the physics of simpler items? Physics is vital for all musical instruments, if it wasn’t; they probably wouldn’t produce the beautiful sounds that they do. One of these instruments is acoustic guitar. By looking at the instrument, it doesn’t look very complicated, but if you delve deeper into its composition, you’ll find that it’s very complicated. Physics takes part in the making of acoustic guitars, all the way to how it produces its beautiful music.
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.
Sound is a type of longitudinal wave that originates as the vibration of a medium (such as a person’s vocal cords or a guitar string) and travels through gases, liquids, and elastic solids as variations of pressure and density. The loudness of a sound perceived by the ear depends on the amplitude of the sound wave and is measured in decibel, while its pitch depends on it frequency measured in hertz, (Shipman-Wilson-Higgins, 2013).
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
Sound is defined as areas of high and low pressure that move outward to form a longitudinal wave. The amplitude and pitch of the sound is dependent on the source and amount of energy produced. Sound is produced by vibrating objects, the vibrations cause disturbances in the surrounding air molecules. When the vibrating object moves outward it causes the air molecules around it to compress and create a high pressure region. As the object moves inward the air molecules expand and create a low pressure region. The high pressure regions are called compressions and the low pressure regions rarefactions. The equation of sound waves is speed = wavelength x frequency or v = f x .
Music is transmitted through sound waves, which are very similar to the sine waves studied in Trigonometry. The differences in the waves result in a different sounds that are transmitted. Vibrating objects travel through a medium (the material that the disturbance is moving through) to create sounds at a given frequency. The frequency is how often the particles vibrate when a wave passes through the medium. The unit that is most used to measure frequencies is the Hertz (Hz) and 1 Hz is equivalent to 1 vibration per second. The frequency affects the pitch of the note that is being played; The higher the frequency the higher the pitch and the lower the frequency the lower the pitch.
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
...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.
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...
peaks of a specific sound wave. The height of the wavelength measures loudness or amplitude.
After reading the first paragraphs you may be wondering why are acoustic waves so important in normal life or “why would I ever need this equation?’ Acoustic waves are important because sound is all around us. When you think of the word “acoustic” a few words may come to mind. Guitars, stereos, and many other items involve acoustic waves. For example, when the bass on a stereo is turned all the way up you m...