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Transcript
Gebze Technical University
Department of Architecture
MAT119
Asst. Prof. Ferhat PAKDAMAR
(Civil Engineer)
N Blok 1-17
[email protected]
Fall – 2016_2017
Week 2
Osmanlı Geometri
- Üç dılı birbirine müsavi müselleslerin irtifaını
nasıl bulurlar.
- Dılın murabbaından, dılın nısfının murabbaını
nakşeder, kök murabbaını alırsın.
-Kaim zaviyeli müselleste, bir kaim zaviyenin
karşısındaki kaim dılın kaim vetere nispetine o
hadde zaviyenin nesi derler
- Ceybi derler
Recall
Trigonometry
Recall Trigonometry
RADYAN
Necessity of Geometry
If you don’t want to yaw from your route, you need geometry!
C
B
A
History of Geometry
Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a
branch of mathematics concerned 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
concerning lengths, areas, and volumes, with elements of formal mathematical science
emerging in the West as early as Thales(6th Century BC).
By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—
Euclidean geometry—set a standard for many centuries to follow. Archimedes developed
ingenious techniques for calculating areas and volumes, in many ways anticipating
modern integral calculus.
The field of astronomy, especially as it relates to mapping the positions of stars and planets on
the celestial sphere and describing the relationship between movements of celestial bodies,
served as an important source of geometric problems during the next one and a half millennia.
Basics of Geometry
- Points, Lines & Planes
- Segments, Rays & Lines
- Distance Between Points
- Distance Formula in “n” Dimensions
- Angles Types of Angles
Types, methodologies and terminologies of Geometry
Absolute geometry
Affine geometry
Algebraic geometry
Analytic geometry
Archimedes' use of
infinitesimals
Birational geometry
Complex geometry
Combinatorial geometry
Computational geometry
Conformal geometry
Constructive solid
geometry
Contact geometry
Convex geometry
Descriptive geometry
Differential geometry
Digital geometry
Discrete geometry
Distance geometry
Elliptic geometry
Enumerative geometry
Epipolar geometry
Finite geometry
Fractal geometry
Geometry of numbers
Hyperbolic geometry
Incidence geometry
Information geometry
Integral geometry
Inversive geometry
Inversive ring geometry
Klein geometry
Lie sphere geometry
Non-Euclidean geometry
Numerical geometry
Ordered geometry
Parabolic geometry
Plane geometry
Projective geometry
Quantum geometry
Reticular geometry
Riemannian geometry
Ruppeiner geometry
Spherical geometry
Symplectic geometry
Synthetic geometry
Systolic geometry
Taxicab geometry
Toric geometry
Transformation geometry
Tropical geometry
…
Fractal Geometry
1 A geometric figure that appears irregular at all scales of length, e.g. a fern
2 A geometric figure which has a Hausdorff dimension which is greater than its topological dimension
3 Having the form of a fractal
4
A mathematically generated pattern that is endlessly complex Fractal patterns often resemble natural
phenomena in the way they repeat elements with slight variations each time
A kind of image that is defined recursively, so that each part of the image is a smaller version of the
whole
6 A fractal is a shape where self-similarity dimension is greater than topological dimension
5
A 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
7
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 realistic
groups that have broken dimensions so that each one looks like an exact copy of the second (like the
8 Mandelbrot group in Mathematics); (In Computers) geometric shapes that have interesting contour
lines
9
A geometric figure that repeats itself under several levels of magnification, a shape that appears
irregular at all scales of length, e.g. a fern
10
A 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 scales
Fractal Geometry
Every fractal is a pattern but every pattern is not a fractal
A pattern can be a fractal with these rules
1- Pattern must be scaled
2- Previous form must be contained
3- Must proceed according to a specific rule
Pattern?
Fractal?
It is a pattern.
Because
Next shape can be predicted
Not a fractal
Because
Shape is not scaled
Fractal Geometry
Every fractal is a pattern but every pattern is not a fractal
A pattern can be a fractal with these rules
1- Pattern must be scaled
2- Previous form must be contained
3- Must proceed according to a specific rule
Pattern?
Fractal?
It is a pattern.
Because
Next shape can be predicted
Not a fractal
Because
Next shape is not encapsulate the previous
Fractal Geometry
Every fractal is a pattern but every pattern is not a fractal
A pattern can be a fractal with these rules
1- Pattern must be scaled
2- Previous form must be contained
3- Must proceed according to a specific rule
Pattern?
Fractal?
It is a pattern and a fractal
Fractal Geometry
Fractal Geometry
Fractal Geometry
Fractal Geometry
Dimension of a Fractal (Hausdorff)
log 𝑁
log 𝜖

−𝐷 =

D: Dimension of a fractal

N: Number of repetitions (total)

𝜖: Scaling factor

What does D describe?
Sample Fractal Figures
These figures are very
important for midterm
exam and Homeworks !
Basics of Geometry
- Points, Lines & Planes
- Segments, Rays & Lines
- Distance Between Points
- Distance Formula in “n” Dimensions
- Angles Types of Angles