Gebze Institute of Technology Department of Architecture MAT119

MAT119Asst. Prof. Ferhat PAKDAMAR (Civil Engineer)M Blok - M106 pakdamar@gtu.edu.tr

Gebze Technical UniversityDepartment of ArchitectureFall 2015_2016

Week 2

Basics of GeometryPoints, Lines & PlanesSegments, Rays & LinesDistance Between Points Distance Formula in n DimensionsAngles Types of AnglesNecessity of GeometryIf you dont want to yaw from your route, you need geometry!BACHistory of GeometryGeometry(from theAncientGreek:;geo-"earth",-metron"measurement") is a branch ofmathematicsconcerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

Geometry arose independently in a number of early cultures as a body of practical knowledge concerninglengths,areas, andvolumes, with elements of formal mathematical science emerging in the West as early asThales(6th Century BC).

By the 3rd century BC, geometry was put into anaxiomatic formbyEuclid, whose treatmentEuclidean geometryset a standard for many centuries to follow.Archimedesdeveloped ingenious techniques for calculating areas and volumes, in many ways anticipating modernintegral calculus.

The field ofastronomy, especially as it relates to mapping the positions ofstarsandplanetson thecelestial sphereand describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia.Types, methodologies and terminologies of GeometryAbsolute geometryAffine geometryAlgebraic geometryAnalytic geometryArchimedes' use of infinitesimalsBirational geometryComplex geometryCombinatorial geometryComputational geometryConformal geometryConstructive solid geometryContact geometryConvex geometryDescriptive geometryDifferential geometryDigital geometryDiscrete geometryDistance geometryElliptic geometryEnumerative geometryEpipolar geometryFinite geometryFractal geometryGeometry of numbersHyperbolic geometryIncidence geometryInformation geometryIntegral geometryInversive geometryInversive ring geometryKlein geometryLie sphere geometryNon-Euclidean geometryNumerical geometryOrdered geometryParabolic geometryPlane geometryProjective geometryQuantum geometryReticular geometryRiemannian geometryRuppeiner geometrySpherical geometrySymplectic geometrySynthetic geometrySystolic geometryTaxicab geometryToric geometryTransformation geometryTropical geometry

Fractal Geometry1A geometric figure that appears irregular at all scales of length, e.g. a fern2A geometric figure which has a Hausdorff dimension which is greater than its topological dimension3Having the form of a fractal4A mathematically generated pattern that is endlessly complex Fractal patterns often resemble natural phenomena in the way they repeat elements with slight variations each time5A kind of image that is defined recursively, so that each part of the image is a smaller version of the whole6A fractal is a shape where self-similarity dimension is greater than topological dimension7A geometric entity characterized by self-similarity (see figure 2): the whole entity is similar to a smaller portion of itself, but has a higher level of recursion (see recursion) Therefore, it can usually be represented by a recursive definition When using a fractal to represent a physical object, some degree of randomness is usually added to make the image more realistic8groups that have broken dimensions so that each one looks like an exact copy of the second (like the Mandelbrot group in Mathematics); (In Computers) geometric shapes that have interesting contour lines9A geometric figure that repeats itself under several levels of magnification, a shape that appears irregular at all scales of length, e.g. a fern10A geometric figure, built up from a simple shape, by generating the same or similar changes on successively smaller scales; it shows self-similarity on all scalesFractal GeometryEvery fractal is a pattern but every pattern is not a fractalA pattern can be a fractal with these rules

1- Pattern must be scaled 2- Previous form must be contained3- Must proceed according to a specific rule

It is a pattern.BecauseNext shape can be predicted

Not a fractalBecauseShape is not scaledPattern?Fractal?

Fractal Geometry

Every fractal is a pattern but every pattern is not a fractalA pattern can be a fractal with these rules

1- Pattern must be scaled 2- Previous form must be contained3- Must proceed according to a specific rule It is a pattern.BecauseNext shape can be predicted

Not a fractalBecauseNext shape is not encapsulate the previousPattern?Fractal?

Fractal Geometry

Every fractal is a pattern but every pattern is not a fractalA pattern can be a fractal with these rules

1- Pattern must be scaled 2- Previous form must be contained3- Must proceed according to a specific rule It is a pattern and a fractalPattern?Fractal?

Fractal Geometry

Fractal Geometry

Fractal Geometry

Fractal Geometry

Basics of GeometryPoints, Lines & PlanesSegments, Rays & LinesDistance Between Points Distance Formula in n DimensionsAngles Types of AnglesRecallTrigonometry

Recall Trigonometry

RADYAN