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GEMATMW > Mathematics in Natural Sciences and Arts > Flashcards

Mathematics in Natural Sciences and Arts Flashcards

1
Q

A pattern has symmetry if there is an ____ of the plane that preserves the pattern

A

isometry

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2
Q

process which shifts points of the plane to possibly new locations in the plane

A

Transformation

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3
Q

types of transformation

A

Translation, Reflection, Rotation, Dilation

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4
Q

mapping that preserves distance (and therefore shape)

A

Isometry of the plane

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5
Q

Four types of plane isometries

A

Translation
Reflection
Rotation
Glide reflection

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6
Q

moves a shape in a given direction by sliding it up, down, sideways, or diagonally.

A

translation

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7
Q

can be thought of as getting a mirror image.

A

reflection

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8
Q

has a point about which the rotation is made and an angle that says how far to rotate.

A

rotation

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9
Q

is a transformation which changes the size of an object.

A

dilation

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10
Q

transformations which leave the dimensions of the object and its image unchanged

A

Isometries
transformations which leave the dimensions of the object and its image unchanged

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11
Q

is a combination of two transformations: a reflection and a translation.

A

Glide Reflection

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12
Q

figure with at least one non-trivial symmetry.

A

design

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13
Q

design that has a translation symmetry

A

pattern

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14
Q

has symmetry if there is an isometry of the plane that preserves it

A

plane pattern

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15
Q

TYPES OF SYMMETRIC PATTERNS

A

ROSETTE PATTERN
FRIEZE PATTERN
WALLPAPER PATTERN

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16
Q

finite groups of isometries which can contain just rotations and reflections

A

Rosette patterns

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17
Q

Two types of rosette patterns

A

Cyclic rosettes
Dihedral rosettes

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18
Q

has n-fold rotational symmetry and no reflectional symmetry

A

Cyclic rosettes

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19
Q

has n-fold rotational symmetry and reflectional symmetry

A

Dihedral rosettes

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20
Q

is an infinitely long strip imprinted with a design given by a repeating motif.

A

Frieze Pattern

21
Q

This pattern is an infinitely long strip imprinted with a design given by a repeating pattern motif.

translational symmetry in one direction

at least the basic unit and a copy of it by translation.

A

Frieze Pattern

22
Q

We imagine that they go on to infinity in both directions or wrap around.

A

Frieze Pattern

23
Q

Frieze Groups

A

hop
step
half turn
sidle
spinning hop
jump
spinning jump

24
Q

hop

A

Only Translational Symmetry

25
Q

STEP

A

Only Translational Symmetry

26
Q

sidle

A

Translation and Vertical Reflection Symmetries

27
Q

spinning hop

A

Translation and Vertical Reflection Symmetries

28
Q

spinning hop

A

Translation, Rotation (Half-turn) and Glide Reflection Symmetries

29
Q

JUMP

A

Translation and Horizontal Reflection Symmetries

30
Q

SPINNING JUMP

A

Translation, Horizontal and Vertical Reflection, Rotation Symmetries

31
Q

covers the plane and can be mapped onto itself by translations in more than one direction.
it must have at least the basic unit, one copy by translation, and a copy of these two by translation in a second direction. That is, there must be at least two rows, each one at least two units long.

A

wallpaper pattern

32
Q

smallest region of the plane such that the set of its images under translations of the pattern generates the whole pattern

A

Basic unit

33
Q

the smallest region in the basic unit whose images under the full symmetry group of the pattern cover the plane

A

Fundamental region

34
Q

subset of a fundamental region which has no symmetry but which generates the whole pattern under the symmetry group of the pattern

A

Motif

35
Q

Patterns in Nature
is a never-ending pattern.
are infinitely complex patterns that are self-similar across different scales.
These are created by repeating a simple process over and over in an ongoing feedback loop.

A

Fractals

36
Q

Involves symmetry of magnification (called dilation)
It is a shape that you could zoom in on a part of it an infinite number of times and it would still look the same

A

Fractals

37
Q

The word fractal was coined in 1980 by the Belgian mathematician

A

Benoit Mandelbrot

38
Q

Self-similar shapes have the property of not being one-dimensional, or two- dimensional, or even three-dimensional, but are instead of fractional dimension.

A

Fractals

39
Q

Famous Fractals

A

Koch Snowflake
Sierpinski Triangle
Mandelbrot Set
Julia Set

40
Q

The Fibonacci sequence was invented by the Italian

A

Leonardo Pisano Bigollo

41
Q

was the outcome of a mathematical problem about rabbit breeding

A

Fibonacci sequence

42
Q

The first two numbers in the are 1 and 1, and each subsequent number is the sum of the previous two.

A

Fibonacci sequence

43
Q

The sequence Fn is of Fibonacci numbers is defined by the recurrence relation

A

Fn= Fn-1 + Fn-2,

44
Q

GOLDEN RATIO

A

1: 1.61803

45
Q

Find the nth Fibonacci number
without using the recursion
formula called

A

BINET’S FORMULA

46
Q

this fractal property refers to the appearance remaining the same under magnification

A

Self Similarity

47
Q

this fractal property refers to how it is not limited to integer dimensions and can have fractions

A

Fractional Dimension

48
Q

this fractal property refers to how it is formed by repeating a process

A

Formation by iteration

GEMATMW (3 decks)

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