Why is 2-D geometry important?
955988
Well, almost every engineered product such as cars and bridges and rockets and even iPods require drawings to show how they're going to be made.  Even for 3D objects, those drawings are still largely 2D, and 2D geometry is indispensible in the development of those 2D drawings.  This is true even for computer workstations that work with 3D representation of products, because all the cross sections are still 2D, and engineers need those cross sections to help with the analysis.
837822
I would have asked the same question when I was in high school, but now that I am in college I see a real use for it. Engineers, mathematicians, scientists, masons, and carpenters use geometry all the time. It's a lot more useful that you think.
348301
'Flatland' must be a very exciting story. I wish I have read it.\n--a 2d world surely is a more interesting place compared to a 1d world. Roughly speaking, the added options would be 'left', 'right', 'up', 'down', which are far more entertaining than the 'forward' and 'backward' of the 1d space. \n-- the 3d world easily beats a 2d world and needs few comments. we get 'depth', just to name one thing we would miss in the 2d world. there are many others, no doubt.\n-- a 4d world would be an awesome place. before else, the question is what do we put on the 4th dimension. whatever would be it must have the dimension of space. so it can be time times velocity. let's say we neglect what velocity would mean and just focus on time. now things are going to heat up. what can we do with time, if we are able to use it just as we use the ordinary space? surely we would like to move forward and backward in time. still, I would not think we would be able to see .... ohhh wait a second!! this is really cool! I think I just got your idea of seeing 'inside and out, of every object and event in the entire univers'. yes, it must be this, because we would be able to travel in time, along the 4th direction, and we would be able to see things as they evolve; suddenly appearing in what was before an apparently empty space, turning into real objects and dissapear eventually. hehe I like it
274232
Euclidean geometry is the study of geometry as defined by the axioms of Euclid. You can find those axioms here: http://en.wikipedia.org/wiki/Euclidean_Geometry .\nIt was noted by Klein that Euclidean geometry was actually just the study of one particular group of symmetries. His Erlangen program of 1872 revolutionized geometry by encouraging the stuidy of non-Euclidean geometries by simply studying the geometries produced by any arbitrary group.\nGauss and Riemann realized that while Euclidean geometry did not deal with a fundamental unit of length, the study of geometry on curved surfaces did introduce fundamental units of length (this was later applied in general relativity). The geometry on these surfaces, such as the sphere where parallel lines can meet at either 2 points or none, definitely did not satisfy Euclid's axioms, so they were also non-Euclidean. The Riemann sphere's relation to the complex plane made this geometry very useful. The projective plane is a non-Euclidean geometry that has found many applications as well. \nBasically, non-Euclidean geometry pops up whenever a surface (or an arbitrary smooth manifold) has non-zero Gaussian curvature. Things can get pretty wild in more than 2 dimensions. You can read about more non-Euclidean geometries oand their applications on this page: http://en.wikipedia.org/wiki/Non-euclidean_geometries
860061
It depends on if you a working in two dimensions or three and if you are working in Euclidean geometry or non-Euclidean geometry.\n\nIn Euclidean plane geometry in two dimensions if the two lines are perpendicular to the same line through two distinct points on that line, they will be parallel.\n\nIf you are in three dimensions  going through two distinct points, they could be parallel if lying in the same plane or not if not in the same plane.\n\nIn non-Euclidean elliptic geometry on the surface of a sphere they would always intersect like longitude lines as they are perpendicular to the equator or latitude lines at different points and all intersect at the north and south poles. \nor \nIn non-Euclidean hyperbolic geometry on the surface of a hyperboloid they would be parallel using an analogy to latitude and longitude lines.
120816
Mainly in architecture (though computers do most of the work these days...).\nAlso, I have a friend who is an artist, and when she creates 3d work (sculptures, interactive art) she uses trigonometry and extremums to decide certain things (the best angle for a part of the work, how to use minimum material when she doesn't have a big budget etc.).\n\nBesides, it's pretty fun, if you like math.
807333
Imaginary numbers, actually complex numbers, are enormously useful in many areas of engineering.  One example is in trigonometry where difficult trigonometric identities become almost trivial to derive.  Another is in analyzing circuits.  Using imaginary numbers allows Ohm's law to be applied to passive (resistors, capacitors and inductors) AC circuits.  Euler's identity allows cubic equations to be solved analytically.  Applying the concept of graphical representation of 2-D vectors provides very powerful tools for the solution of 2-D kinematics problems and difficult geometry problems.\n\nI am a research engineer and I use complex numbers all the time in my work.  It is by far one of the most useful tools that I have to solve difficult problems.
127169
When we are trying to understand many things in the real world, like the design of machinery, or electrical systems, or the weather, we end up with many variables. Examining the system in the space of its variables is very useful; that is, one assigns a dimension to each variable, and draws pictures and analyzes the surfaces and shapes in the space of variables. If there are more than 3 variables, which there usually are, one is in a very high dimensional space.\n\nYou could not solve or analyze very many real world problems without using high dimensional analyses, such as geometry, topology, etc.
1103163
no, two- dimensional objects aren't real in that sense... even a sheet of paper (as close to 2-d as you can get in the physical world) is really 3 dimensional and does have wieght and mass.
570385
Algebra is needed for SO many things! Recipes, budgeting you name it. It teaches problem solving. Geometry is good to shape your mind to think logically. I actually took geometry twice out of my own free will because I loved proving the angles were equal or whatever. My mind was never so sharp. Trig, well I actually haven't found a use for that yet....
1072143
i just got done with geometry and i hated it i'm more of an algebra person.  but to survive geometry you have to take notes and study!  there are so many formulas its not even funny! ugh shapes we're not my best thing in the world. but ask questions when you don't understand because when you don't understand one thing you won't understand a lot of things that go with it.  i hate geometry proofs!!!!!!!!!   check ya later ♥
1044107
None.  That is the point.  If they exist is not proved, and hypothetical tests to see if they exist are flawed.  \n\nOn your other question of visualizing multiple dimensions, I have a few hints.  Have you read Flatland yet?  Start there, its good, and not overly mathematical.  Multiple dimensions can mean different things to different scientists.  We use them all the time for their mathematical properties.  But mathematics is just a description, the real world is something observable.  In terms of observation, you would not be able to detect a separate dimension by any means pedantic.  We may be able to prove that physical space has more than three dimensions, but that does not mean that those other dimensions are physical space.  Just like time.  Some particle interactions appear to go backward in time briefly.  What if there is more than one time dimension?  \n\nSo the classic three dimensional physical picture of multiple dimensions will always be flawed by the observers expectation of the properties of the unseen dimension.  Which is why we describe those dimensions using 'imaginary' constructions.  Hope this helps a bit.
474415
Because of the ratio of area inside a shape compared to the outline isnt the most efficient (outline 4a cf area a^2) where a is the length of side for the square (rationalises to 4:a)\n\none would find other shapes to be more efficient like hexagons, with circles having the best ration between area and circumference (PI*d cf (PI*d^2)/2) which rationalises to 1:d/2 where d is the diameter (width) of the circle.\n\nthe circle will contain the maximum area within for the minimum circumference thus rendering it cheaper if you are say making a box for a known quantity of goods.\n\nother than that the answer can depend on what the final calculations are to be used for
524483
mikii, that's rude, though vinyl is rude and opportunistic as well. shut up if u dont have any answer.\n\nEuclide?\n\nAnyway, plane geometry is also called Euclidean geometry. this is a geometry where 2 parallel line never ever intercept and triangles alway have 180 degree of internal angles. u can imagine it as geometry on a flat plane as we know it. As for spherical geometry, 2 parallel lines do cross and the inside angle of a triangle is larger than 180 degree. how much larger depends on the curvature of the plane and the area of the triangle. U can imagine this geometry as one of the surface of a sphere. Spherical geometry is also called elliptic geometry.\n\nAnother popular geometry is called a hyperbolic. it's a geomtry of the pseudosphere. it's triangles have inside angles of less than 180 degree and 2 parallel line never intercept.
886143
Hi. Euclid did a lot more than that. : http://www.answers.com/topic/euclidean-geometry-1\nGetting folks to think rationally about geometry was a great contribution.
930907
Very good to see that you sincerely want to develop interest in it and have better knowledge.  Geometry at your level fortunately is very visual (since it is mostly 2-d).  The following link has all the Euclid's Elements along with some cool interactive software.  You only need to go through a small part of it..Just play with the software (drag red dots around etc) and develop a feel for geometry first. At first, skip all proofs; just try to understand the statements (postulates, propositions, etc).  That might get you hooked on geometry.  Once you get a good feel for geometry, mensuration falls into place easily.\n\nhttp://aleph0.clarku.edu/~djoyce/java/elements/elements.html\n\nAlso play with some simple ideas: always think of a parallelogram as two triangles joined. Likewise, any polygon as a bunch of triangles.  Visualize a complex object in terms of simpler, more familiar "building blocks".  \n\nPlay string 'games': take a piece of string,  tie the ends together to form a loop.  Lay it flat on a table.  You can take a few match sticks and push out the string with the sticks tight so that the string looks like a straight line between any two sticks. You can make any number of polygons (triangles with 3 sticks, rectangles, squares, trapezoids, etc 4 sticks and so on, all having the same perimeter. Think of maximizing the area inside the string with 3 sticks: what do you notice about the angles, lengths of sides? How about with 4 sticks? And so on. \n\nRemember, develop visual intuition first. But pictures are not proof! So, you have develop the ability to make precise argumens using axioms and previously proved statements as well.  But that can come later.  Understanding and discovering facts is much more important than proofs (do not take this as advice to skip proofs, though). \n\nOne more thing: after having read a proof, sometimes you feel that you followed every step, but yet you don't "get it".  If that is the case, you have not really understood it.  You have to get to a point where a proof is not ten impossible-to-memorize steps each of which you understand.  And with practice, you can!  Think of biking: A robot may need to 'perform many steps' such as holding the bike, lifting the left foot, resting on a pedal...  Perhaps we learn the first time like that; but at some point it becomes so natural that we stop thinking about the individual steps and concentrate on higher goals.  Same thing with math.\n\nYou learn math primarily by doing. Good teachers, books, and software provide an immense boost in the process -- but they are all secondary to your own involvement.\n\nHope that helps and stay with it!
476516
The Geometry of Perspective Drawing on the Computer\nhttp://www.math.utah.edu/~treiberg/Perspect/Perspect.htm