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FOURTH GRADE / SYMMETRY 4.2 / TRANSLATIONAL SYMMETRY

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH .
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" FOR YOUR SIZE.

  
WE WILL SEE THREE TYPES of SYMMETRIES:

 ♦ 
1- SYMMETRIES with respect to A LINE or PLANE .
 ♦ 2- TRANSLATIONAL SYMMETRY .
 ♦ 3- ROTATIONAL SYMMETRY .


THEY ARE REALLY of USE .
y
1. Why can we easily guess the shape of the 3rd airplane ?

!
In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2



a) because of continuity
b) because of symmetry
c) because of repetition
d) because of translational symmetry


Because thre is a nearly perfect translational symmetry in the figure.
y
2. What type of symmetry may we be having in the figure below ?

!
In case of need for tips get the cursor on "y" or for help click at "ry1/2" .

RY1 RY2



a) symm. with respect to a plane
b) symm. with respect to a line
c) symm. with respect to a point
d) symm. with respect to a shape
Look at the ry1.
3. Which of the following figures may have overlapping, without rotation just under translations overlapping, symmetric forms both with respect to a line and a point ?

!
a) I and II
b) I and III
c) I, II and III
d) all
The regular pentagon may have, be careful . Put a mirror below the base of it parallel to base and ... .
4. How many axes of translational symmetry does the following figure have ?

!


a) 0
b) ∞
c) 1
d) 2
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation. Discrete translational symmetry is invariance under discrete (a certain amount of distance) translation .
The figure is assumed to be infinite, two dimensional, plane with the pattern repeating .
5. How many axes of translational symmetry does the following figure have ?

!


a) ∞
b) 4
c) 2
d) 1
One vertical translation is easily seen, is another (hybrid) translation possible ?
6. Where may there be an application of discrete translational symmetry, invariance of the figures and forms after a discrete space interval as in question four or five ?

a) movie making
b) art
c) music
d) wallpaper production, newspaper etc. printing
Print a definite part via a drum and repeat it as the paper rolls .
ClickN KIDS Beginning Reading & Spelling Programs Teacher Created Resources ClickN KIDS Beginning Reading & Spelling Programs
7. How big is the transverse (any bird to upper next) translational symmetry interval in the following figure ?

!
a) 2 cm.
b) 5 cm.
c) 4 cm.
d) 3 cm.
Do never forget 3-4-5 triangle .
8. The letter frequencies for English are given below. The most often used letter is "e" with an approximate frequency of 13 % .
Assuming
- a line of lengt of 200 characters
- no space character and
- character to character distance of 1/2 cm.
and covering all characters except for e, what would the average translational symmetry interval of the script be ?

!
a) 9 cm.
b) 8 cm.
c) 6 cm.
d) 4 cm.
The text will be something like " ♦♦e♦♦♦♦♦e♦ "
Within 200 character there should be, in the mean, 26 "e". The avearge number of characters between two "e" should be 200/26 = 8 (approx.) .
9. Average word length in English is accepted as 5. This means every 6th character, in the mean, is supposed to be a space.
How would this modify the result of the previous question ?

a) 4 cm. will decrease to 3 cm.
b) will increase to 6 cm.
c) will be almost the same
d) 4 cm. will increase to 5 cm.


200 characters would require 40 spaces . In 240 characters there would be 26 "e" . The mean distance between two "e" would be 120/26 cm. = 5 cm. (rounded to the nearest natural number) .
10. When we say that the figure below has translational symmetry we mean "shifting this figure by a certain amount we get the same figure".
What is a necessity for this claim to be true ?

!


a) infinite dimensions
b) a repetition of finite amount
c) the figure should be infinite at least in 1 direction
d) no need for extras
For a finite figure we can not have this property .
11. We claim that the figure below has infinitely many axes of translational symmetry. What does this imply ?

!


a) infinite in both directions
b) is in 3 dimensions
c) time is the 4th dimension
d) nothing
it should be on the whole plane .
12. How many symmetries are possible for the following figure ?

!


a) 0
b) 2
c) 4
d) infinitely many
Infinitely many in two dimensions .
13. Are both the figure of question 10 and the following one equivalent as far as the number of translational symmery axes are concerned ?

!


a) yes, they both have infinite
b) no, the former's axes are denser
c) they have equal number of axes
d) the numbers are uncountable
The first one has more elements per unit area and ... .
14. The following figure has infinitely many translational symmetry axes and an infinite amount of symmetries do exist. Translation along ... axes are enough to generate all of these symmetries .

!


a) 2 vertical
b) 2 transverse
c) 1 vertical and 1 horizontal
d) 2 horizontal
Two translations, shift by 1 in + and - 45° , gives us all. They are also called the (group) generators .
15. Regarding the figure of question 13 the following data are given. How many of the following operations create(s) a symmetry?

!


I- an horizontal shift by 98 mm.
II- a vertical shift by 2x8.6 cm.
III- a vertical shift by 1.5x98 mm. following an horizontal shift by 1.5x86 mm.
IV- an horizontal shift by 0.49 m. following a vertical shift by 0.98 m.

a) 3
b) 4
c) 2
d) 1
Being quantitative is of certain use . Be carefull about the last two.
16. Regarding the figure of question 13 and considering a translation as an operation, which operation gives us back the figure we started ?

a) a shift of magnitude 0
b) any translation
c) a shift by one square
d) no such operation is possible
The image should stay still, no other way .
17. Is it always possible to turn back to our old figure we started with, after having made a definite number of translations ?

a) no, we may have gone too far away
b) no
c) possibly
d) yes, the shortest way not necessarily being over the inverses of all translations
The inverse translation, that brings us back to our old cannon, from the resulting figure should be made .
18. How should we name the totality of all translations that brings us back to our original figure ?

a) identity
b) unity
c) equivalent
d) zero
Any number of the translations may be taken as a forward translation and the remainings its inverse .
19. What is the form of the unit cell in the following tiling ?

!


a) a square
b) a cube
c) an hexagon
d) an octagon
There are defects but the maim cell has the form of a(n) ...gon .
20. In the figure below we see three cubic forms alltogether having one dimesional translational symmetry. Which choice gives the right names of the cubic forms

!


a) all cubic
b) cubic, body centered cubic and face centered cubic
c) all rectangular prisms
d) all tetrahedral
If one of the spheres is in the center of the cube the form is called body centered cubic .
21. Does the places of rest of the traveler of question-21 of the previous test have translational symmetry ?

!


a) yes
b) possibly
c) perhaps
d) no
The travel is linear and begins from the crossing point of the two axes.
y
22. Below in the figure there are 4 pulses that have one axis of translational symmetry. What will be the mean uncovered area per pulse if we approximate the first one as the sum of the following three ?
Note: The length of the first pulse is 15 units and its height is 8 units. Give the uncovered area as ... unit2 .

In case of need for tips get the cursor on "y" or for help click at "ry1/2".

RY1 RY2



a) 20
b) 40
c) 60
d) 50


We want to introduce you the preliminaries of Furier Transform which is very important. We asked you the area because it is simpler.
The sum of the last three figures is given in ry1.
23. Which of the followings have/has translational symmetry (discrete or continuous) ?

I- a line
II- a line segment
III- a ray
IV- a plane

a) I and IV
b) I and II
c) I, II and IV
d) all
If bounded at any side, no symmetry .
24. Which of the followings have/has translational symmetry (discrete or continuous) ?

I- natural number's axis
II- integer's axis
III- grid of natural number's axis
IV- grid of integers

a) II
b) I
c) IV
d) III
Integers include both positive and negative natural number (and 0) .
25. How many axis of translational symmetry does the following figure have ?

!


a) 0
b) 1
c) 2
d) ∞
Only vertical shifts create symmetry .
This is the feedback!

THANK YOU FOR YOUR VISIT; HOPE WE COULD BE OF USE.
FOR MUTUAL BENEFIT YOU MAY USE OUR FORUM.
© 2011 , İsmail GERMAN. ALL RIGHTS RESERVED
PLEASE PREFER GIVING LINK IF THAT SUFFICES

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