When it comes to Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry there are many similarities and differences among them. For example, what may be true for Euclidean Geometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where something is true for one or two geometries but not the other geometry. However, sometimes a property is true for all three geometries. These points bring us to the purpose of this paper. This paper is an opportunity for me to demonstrate my growing understanding about Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry.
The first issue that I will focus on is the definition of a straight line on all of these surfaces. For a Euclidean plane the definition of a “straight line” is a line that can be traced by a point that travels at a constant direction. When I say constant direction I mean that any portion of this line can move along the rest of this line without leaving it. In other words, a “straight line” is a line with zero curvature or zero deviation. Zero curvature can be determined by using the following symmetries. These symmetries include: reflection-in-the-line symmetry, reflection-perpendicular-to-the-line symmetry, half-turn symmetry, rigid-motion-along-itself symmetry, central symmetry or point symmetry, and similarity or self-similarity “quasi symmetry.” So, if a line on a Euclidean plane satisfies all of the above conditions we can say it is a straight line. I have included my homework assignment of my definition of a straight line for a Euclidean plane so that one can see why I have stated this to be my definition. My definition for a straight line on a sphere is very similar to that on a Euclidean Plane with a few minor adjustments. My definition of a straight line on a sphere is one that satisfies the following Symmetries. These symmetries include: reflection-through-itself symmetry, reflection-perpendicular-to-itself symmetry, half-turn symmetry, rigid-motion-along-itself symmetry, and central symmetry. If we find that a line on a sphere satisfies all of the above condition, then that line is straight on a sphere. I have included my homework assignment for straightness on a sphere so that one can see why a straight line on a sphere must satisfy these conditions. Finally, I need to give my definition of a straight line on a hyperbolic...
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...h other along a third line, l. Then to consider the geometric figure that is formed by the three lines and look for the symmetries of that geometric figure. Then we were asked what we could say about the lines r and r’. I have provided my notes that include an outline to this proof for all three surfaces so that one can see the conclusions that we made as a class. We found that on a Euclidean plane parallel transported lines do not intersect and are equidistant. For a hyperbolic plane we found that parallel transported lines diverge in both directions. Finally for a sphere we found that parallel transported lines always intersect.
Using all the above material, we can see that there are many different similarities and differences when looking at a Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry. Using my artifacts will help one understand many of my conclusions about these three surfaces. This essay was an excellent opportunity to reflect on my growing understanding of these three surfaces. I hope you, the reader, can benefit from my conclusions and gain a better understanding of the similarities and differences of these three surfaces.
Upon completion of this task, the students will have photographs of different types of lines, the same lines reproduced on graph paper, the slope of the line, and the equation of the line. They will have at least one page of graphing paper for each line so they can make copies for their entire group and bind them together to use as a resource later in the unit.
He also illustrates principles of design. If you were to place a vertical line on the picture plane the two sides would balance each other out. The painting can also be divided half horizontally by the implied divisional line above the horses head and the sword of the man who St. Dominic has brought back to life. Contour horizontal lines that give the expression that the dead man on the ground is sliding out of the picture plane, and dominate the bottom of the painting. On the top of the picture plane, behind the spectators is the brightest intermediate color, which is red orange that gives the impression of a sunrise.
Abstract geometric shapes are easily seen in “Always.” The subject of geometry is apparent from any angle. This sculpture has a unitary form of a long and large rectangular shape that bends several times in different directions and angles before springing into space. Mass controls the composition of this sculpture; it is a large sculpture with lots of volume. The mass creates a line of movement into space. There is a dynamic action of the geometric shapes extending into space. The sculpture appears unstable and off-balanced as if it is going to tip over.
Study of Geometry gives students the tools to logical reasoning and deductive thinking to solve abstract equations. Geometry is an important mathematical concept to grasp as we use it in our life every day. Geometry is the study of shape- and there are shapes all around us. Examples of geometry in everyday life are- in sport, nature, games and architecture. The game Jenga involves geometry as it is important to keep the stack of tiles at a 90 degrees angle, otherwise the stack of tiles will fall over. Architects use geometry everyday- it is essential when designing buildings- shape, angles and area and perimeter are some of the geometry concepts architects
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.
In this experiment, two vertical lines will be presented to the participants in each trial. There are 130 trials. The line with outward wings will always keep the same length. The line without wings (also known as perceived line in CogLab, Muller-Lyer Illusion, Cengage Learning) will change its length from trial to trial. Participants will be asked to choose which line is longer in each trial. Participants have to complete each trial in a few second in order to make the illusion effect...
“Deriving the Parallax Formula” shows that one way of deriving the parallax formula is to set up a right triangle consisting of Earth, the Sun, and one other star as vertices. The side going from Earth to the Sun can be labeled as “a” and the side from the Sun to the other star can be labeled as “d.” The angle between the other star and Earth can be labeled as “p.
Lines are one of the fundamentals of all drawings. The lines in this drawing represent shape, form, structure, growth, depth, distance, movement and a range of emotions. In “Three Mile Island” Jacquette uses a mixture of horizontal lines to suggest distance and calm, through his use of thick and thin lines he shows delicacy and strength.
Knowing Venus of Willendorf is a sculpture, she has very nice defined lines. She has a nice combination of vertical, horizontal and curved lines. The artist has given her a nice horizontal line crossing across her breast that her arms create that draw you in. From the horizontal line to the vertical line that draws your eye down to look at her genitalia. She also has nice curved lines that form all around her. From the top of her head, to her breast, to the middle of her stomach, that bring your eye to her behind and back to the front of her legs. The artist has created a nice curved lines that surrounds her breast, as well as her stomach and rear hind. Another way to view her is from the side, which gives you a nice sense of her curved lines that you eye follows down in a flow.
The sphere, hanging from a long wire set into the ceiling of the choir, swayed back and forth with isochronal majesty. I knew—but anyone could have sensed it in the magic of that serene breathing—that the period was governed by the square root of the length of the wire and by IT, that number which, however irrational to sublunar minds, through a higher rationality binds the circumference and diameter of all possible circles. The time it took the sphere to swing from end to end was determined by an arcane conspiracy between the most timeless of measures: the singularity of the point of suspension, the duality of the plane’s dimensions, the triadic beginning of ir, the secret quadratic nature of the root, and the unnumbered perfection of the circle
The Gauss-Codazzi equations are fundamental equations in Riemaniann geometry, where they are used in the theory of embedded hypersurfaces in a Euclidean space. The first equation was derived by Gauss in 1828 (Gauss, 1828) and is the basis for Gauss’s “Theorema Egregium”, which states that the Gaussian curvature of a surface is invariant under local isometry. It relates the n-curvature in terms of the intrinsic (n-1)-geometry of the hypersurface. The second equation is named for Delfino Codazzi, although it was derived earlier by Peterson in 1853 (Peterson, 1853), and relates the n-curvature in terms of the extrinsic curvature (often referred to as the second fundamental form) of the hypersurface.
Lines are paths or marks left by moving points and they can be outlines or edges of shapes and forms. Lines have qualities which can help communicate ideas and feelings such as straight or curved, thick or thin, dark or light, and continuous or broken. Implied lines suggest motion or organize an artwork and they are not actually seen, but they are present in the way edges of shapes are lined up.
On first thought, mathematics and art seem to be totally opposite fields of study with absolutely no connections. However, after careful consideration, the great degree of relation between these two subjects is amazing. Mathematics is the central ingredient in many artworks. Through the exploration of many artists and their works, common mathematical themes can be discovered. For instance, the art of tessellations, or tilings, relies on geometry. M.C. Escher used his knowledge of geometry, and mathematics in general, to create his tessellations, some of his most well admired works.
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.
Abstractions from nature are one the important element in mathematics. Mathematics is a universal subject that has connections to many different areas including nature. [IMAGE] [IMAGE] Bibliography: 1. http://users.powernet.co.uk/bearsoft/Maths.html 2. http://weblife.bangor.ac.uk/cyfrif/eng/resources/spirals.htm 3.