The Fibonacci sequence is often defined as {F_n }_(n=1)^∞ containing the numbers 1, 1, 2, 3, 5, 8, 13, 21, and so on [1]. A formula for determining the numbers contained in the set is given by F_1=1 and F_2=1, with the recursive formula F_n=F_(n-1)+F_(n-2). In other words, the formula doesn't start until n=3, and computing elements of the set just involves adding the two previous numbers together to get the next. Using the Fibonacci sequence, it is true that the infinite sequence r_n, {F_n }_(n=1)^∞ {r_n }_(n=1)^∞=F_(n+1)/F_n lim┬(n→∞)〖F_(n+1)/F_n =φ〗 {1,1,2,3,5,8,13,21,…} F_n=F_(n-1)+F_(n-2) F(x)={█(1,x=1@1, x=2@F(x-1)+F(x-2), x≥3)┤ is convergent and ends up converging to φ, the golden ratio [2]. This curious quantity is just a ratio, so what makes it so special? Why is it so mystifying? While the other major constant in mathematics, pi, is a ratio between a circle's circumference and its diameter, phi (φ) considers a rectangle with height, h, and width, w, and forms the following ratio: 1/w=w/(h+w) To get the value for φ, we should assume h=1. Despite the assumption, this does not lose generality for the ratio [2]. Thus, 1/w=w/(1+w) Cross multiplying gives us the following: w^2=1+w Algebraic manipulations yield a quadratic equation in terms of the width, w. w^2-w-1=0 This can easily be solved using the quadratic formula, which gives the result: w=(1±√5)/2 We cannot have a negative width, so the negative answer is not considered. Therefore, the golden ratio is given by: φ=(1+√5)/2 Therefore, φ is a solution to w^2=1+w. In fact, a rectangle with side lengths φ is said to be a golden rectangle, which is a result of the assumption that h=1 [2]. The number phi has even gripped theologians to ... ... middle of paper ... ...unappealing to me. With the awkward angle, the fact that the photograph is not centered correctly makes matters worse. There is no balance in the presentation of this picture in contrast to the other photographs available online. Take the Sacrament of the Last Supper by Salvador Dali for example. It's perspective is perfectly blended together to make a pleasing image to the viewer. Disregarding phi, the simple fact the there is a clear focus makes the image more likeable to me. Plenty of other pieces of artwork tend to follow the same idea by at least balancing the image on either side with action or activity in the center of the painting that is not necessarily the focus. There is a certain point where the landscape or picture "just isn't right." This stands for any piece of artwork or photograph, and is often attributed to the lack of a central focus in my case.
What I see in this piece is peacefulness. Stokes of the paintbrush are perfect to make it look whole. With the sun shinning down making the colors pop out even more. The olive trees glowing in the suns light with the mountains behind it. It is a piece I could look at for a long time with out getting bored. The colors of the piece just make it look so complete. With the lines of the
Once punctum occurs we become fixated on that something that provokes us, a single detail of a photograph that holds our gaze without condescending to mere meaning or beauty.
This is his focus of the painting. The focus can be defined as the main point of a painting, the area
This paper will discuss three specific instances: Le Sacrifice, Psappha, and Metastasis. The first principle that I will discuss is the Golden Section. The Golden Section can be found in art and architecture dating as far back as the Parthenon, as well as different places in nature, such as the nautilus shell. The Golden Section is essentially a proportion that is established by taking a single line and dividing that line into two separate sections of unequal lengths, one quite longer than the other.
Nevertheless, that day followed me, and I tried to understand more about fractals through the resources I already had at my disposal-- through courses I was taking. Sophomore year, through my European History and Architecture courses, I learned about many ancient architectural feats-- Stonehenge, the Pyramids of Giza, the Parthenon, many Gothic Cathedrals, and the Taj Mahal-- and that they all somehow involved the use of the golden ratio. I will come back to how this relates to fractals later in the article, but for now know that each of these buildings use different aspects of their design to form the golden ratio. I was intrigued by the fact that fractals, what seemed to be something only formed by the forces of nature, were being constructed by human hands. Although I wanted badly to find out more, I waited until that summer, when I discovered a YouTube account by the name of Vihart. Vihart’s videos are not tutorials on how to do math, however Vihart’s ramblings about the nature and the concepts of the mathematical world have a lot of educational value, especially on topics that are more complicated to understand then to compute. Her videos on fractal math and their comparability to nature, helped to show me that...
Fractal art is a new-age art that tantalizes the eyes and mind with patterns, shapes, colors, and abstract imagery. Artists have once again found a way to harness the abstractedness of mathematics and integrate it into their work. So where does this new art form of fractal design stem from? The reality is that fractals themselves are relatively young in the mathematical world. Of course since the beginning of art and history and mathematics, self-similar objects have existed and been intriguing to the human mind. However it has only been recently that mathematicians have begun to explain them. So the question is posed, what is a fractal?
The photo has been taken outdoors on a dirt path that is leading to a forest. On the dirt path, there are the three children which make it more pleasing to the eye and also ties the image together. The girl in the middle facing towards the camera is the subject matter this is because she has eye contact with the camera and because of the intense brightness of her dress, which leads you to look straight at her. The middle girl’s hair is left messy even though she is dressed neatly this adds texture and realism to the photo because people can relate and upstand what it feels like, so views now have more of a connection the image. The tone of this image is a tad gloomy because of the darkness of the forest. The contrast in this image is not high because the elements of the picture all hold sameness. This image has a shallow depth of field which is
Overall the artist does make a unified scene in this composition. Birch used these principals of design to make his composition more effective like balance, movement, repetition and unity. The composition seems balanced because most of the subjects in the painting are all equally distributed and proportioned.
this could shift the main focus from the most subject to it unimportant object. it's going to be a light-weight post or a bush for example. Therefore, we propose that you simply do not pay an excessive amount of attention to the most subject of the image. 3 - The frame edge At times, folks cross-check a photograph solely to raise, "Where did their feet go?"
When looking at an art piece such as a landscape oil painting by Albert Bierstadt—American artist who created Yosemite Valley. “In 1859, he traveled westward in the company of a land surveyor of the U.S. government, returning with sketches that would result in numerous finished paintings” (wikipedia). The artist shows incredible attention to detail throughout the landscape. The use of reflections is extremely effective and visually accurate; the reflections in the water—clarity of trees, shrubs, pebbles, and rocks—might be one of the most noticeable features in this piece. The relative brightness creates the warmth felt in this painting. The interposition as well as the allusion of three dimensions provides a high level of depth. Aerial perspective is correctly used, furnishing the effect of distortion—the...
‘Nature abounds with example of mathematical concepts’ (Pappas, 2011, .107). It is interesting how much we see this now we know, regarding the Fibonacci Sequence, which is number pattern where the first number added to itself creates a new number, then adding that previous number to the new number and so on. You will notice how in nature this sequence always adds up to a Fibonacci number, but alas this is no coincidence it is a way in which plants can pack in the most seeds in a small space creating the most efficient way to receive sunlight and catches the most
...on of light and the rays are proportions in the Fibonacci sequence. Fibonacci relationships are found in the periodic table of elements used by chemists. Fibonacci numbers are also used in a Fibonacci formula to predict the distant of the moons from their respective planets. A computer program called BASIC generates Fibonacci ratios. “The output of this program reveals just how rapidly and accurately the Fibonacci ratios approximate the golden proportion” (Garland, 50). Another computer program called LOGO draws a perfect golden spiral. Fibonacci numbers are featured in science and technology.
Named after the Polish mathematician, Waclaw Sierpinski, the Sierpinski Triangle has been the topic of much study since Sierpinski first discovered it in the early twentieth century. Although it appears simple, the Sierpinski Triangle is actually a complex and intriguing fractal. Fractals have been studied since 1905, when the Mandelbrot Set was discovered, and since then have been used in many ways. One important aspect of fractals is their self-similarity, the idea that if you zoom in on any patch of the fractal, you will see an image that is similar to the original. Because of this, fractals are infinitely detailed and have many interesting properties. Fractals also have a practical use: they can be used to measure the length of coastlines. Because fractals are broken into infinitely small, similar pieces, they prove useful when measuring the length of irregularly shaped objects. Fractals also make beautiful art.
Fractal Geometry The world of mathematics usually tends to be thought of as abstract. Complex and imaginary numbers, real numbers, logarithms, functions, some tangible and others imperceivable. But these abstract numbers, simply symbols that conjure an image, a quantity, in our mind, and complex equations, take on a new meaning with fractals - a concrete one. Fractals go from being very simple equations on a piece of paper to colorful, extraordinary images, and most of all, offer an explanation to things. The importance of fractal geometry is that it provides an answer, a comprehension, to nature, the world, and the universe.
The Golden Rectangle is a unique and important shape in mathematics. The Golden Rectangle appears in nature, music, and is often used in art and architecture. Some thing special about the golden rectangle is that the length to the width equals approximately 1.618……