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Answers:wow someone doesn't want to do their homework. If you can't do these you are going to fail miserably. Here are my mole links, teach yourself. http://www.kentchemistry.com/links/MathofChemistryLinks.htm
Answers:Things I'd consider: 1. Look up "ray tracing". That's how light reflects and so on. Lots of "angle of incidence = angle of reflection". 2. I don't think the Golden Mean is particularly relevant, but it's HUGE in more static visual arts like painting, so look it up too. 3. Perspective, in painting and film alike, is all about geometry. Similar triangles for the win! 4. Image compression and so on are pretty sophisticated. But recognizing boundaries of objects certainly comes into play.
Answers:Throughout the world of music, there are patterns that can be explained by math. Math is related to four major topics of music: beat, tone, tune and the composition itself. The types of math that can be used are geometry, transformations, and very simple uses of addition and multiplication. The first genre of music that needs to be explored is the beat of the music for it is the basis of all music. "The story of the connections between math and music necessarily begins with rhythm. Rhythm is the basis upon which music is built, just as the concept of number is the basis of mathematics." (Garland, 6) Rhythm is a way to measure time between beats. Likewise, music is divided into measures for which the length is determined by the time signature. Each measure in a composition is equal and contains the same amount of beats. The time signature is placed at the beginning of each composition and resembles a fraction. The "numerator" tells how many beats are in one measure while the "denominator" tells what kind of note receives one beat. The time signature "four four" is the time signature that gives the notes their names. For instance, if a measure with a "four four" time signature is filled with only one note, it is called a whole note. If it is filled with two notes, each of those notes is called a half note and likewise if it is filled with four notes, they are called quarter notes. Therefore a half note equals of a measure, a quarter note equals of a measure, an eighth note equals 1/8 of a measure and so forth. There are also notes that come between the basic notes. These notes are said to be "dotted". When a note is dotted, it means that it is equal to the note itself plus half of the note. For example, a dotted half note is equal to the length of a half note plus of the length of a half note. + ( of ) = + = Therefore a dotted half note takes up of a measure. The fact that the time signature is like a fraction has other implications. Because in the time signature "four four", 4/4 = 1, the lengths of the notes in one measure must add up to 1. For a time signature of "three four", the quarter note still gets one beat but there are only three beats in each measure. Therefore, the fractions of each note in a measure should add up to . Over all, the mathematic implications in the rhythm of music are easy to see as well as simple in nature. Another subject of music that is related to math is the tone of the notes. Sound travels in waves. The "top" of the wave is called the crest and the "bottom" of the wave is called a trough. From one crest to another or from one trough to another is one cycle. The kind of wave that sound makes is called a transverse wave because the "particles" of the wave travel parallel to the direction of the wave When studying waves, there are four items that come into play: amplitude (A), wavelength (?), frequency ( ), and period (T). The Amplitude is of the vertical distance from the troughs to the crests. The wavelength is equal to the distance from one crest to another, or the distance the wave moves in one cycle. The frequency of the wave is the number of waves that go through a point in one second. Therefore the period is the time it takes for one whole wave to pass through one point and is equal to the reciprocal of the frequency. All these characteristics of the wave modify the tone of the note that is being produced. The Amplitude affects the loudness of the note. A smaller amplitude produces a softer sound while a larger amplitude produces a louder sound. The frequency of the wave affects the pitch of the note. A pitch is how low or high the note sounds. A low frequency produces a low pitch and a high frequency produces a high pitch. A note that has constant frequency and amplitude and is the shape of a sine curve is called a pure or simple tone. When two or more pure tones are combined a resultant tone is produced. To calculate what the resultant wave curve will look like, it is necessary to add the displacement of the two pure tones. The displacement is the distance from the "middle line" of the curve to the points on the curve. Most musical notes or tones are made of more than two component pure tones. These pure tones are called partials and they each have their own frequencies. For a musical tone, the tone that has the smallest pure-tone frequency is called the fundamental and has a fundamental frequency ( ). "In most musical tones, the frequencies of the partials are integer multiples (2 , 3 , 4 , ) of the fundamental frequency. Such a tone is said to be made from its harmonics or harmonic overtones." (Garland, 32) Something unusual happens when two tones of different frequencies are heard simultaneously. The different sound waves create interference and a beat frequency is heard, like a wavering of soft and loud in the sound. The beat frequency is the difference between the two frequencies that are being heard together. Therefore it is expected that two notes of the same frequency, or two notes in unison, would produce no beats. In order to standardize the music world, some set frequency is needed. Today, the standard is that an "A" has a frequency of 440 Hz. Because the frequencies need to be set to something, the tune can be called objective. This concept of set frequencies can be more clearly understood in the study of the organization of the piano keyboard. The keyboard of a piano includes 88 keys both black and white and a pattern that repeats itself every twelve keys. Each repetition has 7 white keys and 5 black keys. The white keys are named A through G and the black keys are named corresponding to the two white keys they are in between. If a black key is one above a "C" it is called a "C#" or C sharp. The "C#" is also one key below a "D" and is therefore also called a "Db" or D flat. All of the black keys subsequently have two names because all are between two different white keys. Notice that there are not black keys in between the "E" and the "F" and the "C" and the "B". A move from one key to the next key up is a half step. For example, there is one half step between C and C#. Because there is no black key between E and F, there is also only one half step between E and F. A whole step consists of two half steps. From C to D is a whole step. Scales are determined and defined by the number of steps between each note. A major scale consists of two whole steps, a half step, three more whole steps, and a final half step. If beginning on an F, the major scale would go as follows: F, G, A, A#, C, D, E, and F (the half steps being from A to A# and from E to F.) A chromatic scale is made up of 13 half steps, playing all the keys in one octave. From one note to the next note of the same name higher on the keyboard is called an octave. From one C to the next is an example of an octave. There are special relationships between notes that are an octave apart. There is also a correlation between math and music when it comes to the frequencies of these notes. The frequencies (in Hertz) of each note are as follows. It is obvious that there is a ratio of 2:1 in the frequencies of two notes and octave apart. There is also a relationship between the frequencies of two notes right next to each other. This can be found by dividing the frequency of a note by the frequency of the note right below it. The frequency of A# is 233.1 Hz and the frequency of A is 220 Hz. Therefore the ratio is 1.059463094. This is a constant ratio throughout all of the half step intervals. When the frequency is graphed against the notes, the graph is an exponential curve. It is interesting that there are 12 half steps in an octave and the number 1.059463094 = 2 1/12. Trudy Garland and Charity Kahn put it best in their book, Math and Music: Harmonious Connections when they said: A and A#, for example, form an interval of a half step. Call the ratio of their frequencies h. Then, A# / A= h and A# = A x h. Because all of the notes have frequencies equal distances apart, the scale is called even-tempered. Music in Asia is often divided into 24 tones while in s