# Capital Letters Have Rotational Symmetry

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Explanation: Geometry Labs Part 5 Punctuation 71 5. Sort the uppercase letters into two groups below: those with inversion and those without. Example conversion:

## Capital Letters Have Rotational Symmetry

5 Questions Most of the activities in this section provide opportunities for students to express their ideas through mathematics and technology—you should be able to create several screen displays from the activities they do in different labs. Maybe you should plan to put together a final show based on the best work from this section. (See the teacher’s notes to Lab 5.5 for ideas on displaying shapes.) Note that the first lab is a prerequisite for all the others. An excellent resource on the mathematics of regularity and on creative notation for regular patterns is Handbook of Regular Patterns: An Introduction to Queries in Two Dimensions by Peter S. Stevens. In addition to explaining the ideas, the book contains thousands of meaningful images from almost every culture in the world. Martin Gardner’s The Ambidextrous Universe is an excellent meditation on balance. It opens with the question, “Why do mirrors rotate left and right, but not up and down?” See page 191 for teacher notes to this section.Geometry Labs Section 5 Symmetry 69 © 1999 Henri Picciotto, www.MathEducationPage.org LAB 5.1 Name(s) Introduction to Symbols 1. These letters are linear: These letters are not : a. Explain the difference. b. Draw a line or parallel lines over each letter in the first group. (In some cases, there are two lines.)2. Show how each of the five capital letters not shown in Problem 1 can be written in two ways: a straight line, or not. 3. Arrange the lowercase letters in two rows below: straight line, not. Linear Symmetric: Not Value: 4. Below are some numbers that have an inverse and some numbers that don’t. Explain the difference.Find an equation of rotation It has no rotation sense70 Section 5 Symmetry Geometry Labs © 1999 Henri Picciotto, www.MathEducationPage.org LAB 5.1 Name(s) Introduction to Questions (continued)5. Arrange the uppercase letters into two lists below: those with inverted commas and those without. Sort the lowercase letters into two groups below: those with a hyphen and those without. Poet Scott Kim has found clever ways to arrange letters so that he can compose meaningful words. Explain the definition of design below. From Inversions, by Scott Kim.8. On a separate sheet of paper, write meaningful words. In this case, the meaning can be interpreted in many ways, using uppercase or lowercase letters. The following list provides possible interpretations. in. Words with horizontal lines are aligned, such as COB. b. Words with vertical alignment lines, such as TOT. (To make this easier, you can write words vertically, with each letter under the previous one.) c. Pairs of words that are mirror images of each other, such as box and pox (using horizontal mirror). d. Words that can be read upside down, like SOS. yes Pairs of words where one word is the other word upside down, like MOM and WOW. favorite. Palindromes, such as RADAR or RACE CAR (aka space).Geometry Labs Section 5 Symmetry 71 © 1999 Henri Picciotto, www.MathEducationPage.org LAB 5.1 Name(s) Introduction to Symmetry (continued) DiscussionA. Also called mirror line projection, mirror reflection, parallelogram, and inversion. Explain why each of these words is appropriate.B. What happens if a rectangular shape is folded along parallel lines?C. For the figures in Problem 4, draw vertical lines on the figures that have parallel lines, and draw the midpoints on those that have inversions. Note that some of the figures have two different shapes.D. The numbers in Problem 4 with inversion symbols include examples of 2-fold, 3-fold, 4-fold, 5-fold, and 6-fold inversions. Classify each number according to the specific type of inversion it is. with him. Explain what it means for the figure to have an “n-fold” rotation of eighty.E. If the figure does not change when you rotate it around a point by 180 °, it has a special type of rotation that is also called half-double symmetry, central shape, and symmetry. Explain why each of these words is appropriate. F. What is the meaning of the word symmetry? Compare it with other students’ definitions and dictionary definitions.G. Which letters have a line and an inflection? And what unites them? H. Some characters are ambiguous and can be written either with or without line alignment. Some can be written either with or without inflection. Draw and explain examples of these ambiguities. • Explore how different shapes reinforce or detract from the same shape.72 Section 5 Symmetry Geometry Labs © 1999 Henri Picciotto, www.MathEducationPage.org LAB 5.2 Name(s) Triangle and Quadrilateral SymmetryEquipment : Dot or graph paper, isometric dot, graph Eighth type of triangles Half-equilateral (HE) Right scalene (RS) M isosceles (AI) Obtuse scalene (OS) Equilateral (EQ) Obtuse isosceles (OI) Right isosceles (RI) Acute scale (AS) Eight types of quadrilateralsSquare SQ): regular Rhombus (RH): all sides are equal to a square (RE): all angles 90˚ Parallelogram (PA): two pairs of parallel linesKite (KI): a pair of different shapes, but not all sides are equal to the Isosceles trapezoid. IT): equal one pair of parallel segments, two pairs of opposite-parallel general trapezoid (GT): equal one pair of parallel segments General quadrilateral (GQ): none of the top 1 Find two examples of each type of figure listed above, triangles and quadrilaterals, on lined paper or graph paper. Label them. 2. On each figure, draw shape lines if any. 3. On each figure, mark the center of the figure if there is a rotation. 4. Fill in the table below by inserting triangles and quadrilaterals in the appropriate spaces (some spaces may be empty, others may have more than one entry). Shape-inversion of a line No doubler 2-fold 3-milling 4-None line Two lines Three lines Four lines Four linesGeometry Labs Section 5 Symmetry 73. Is there a relationship between the numbers of the lines of a figure and the state of rotation? Explain.B. For each blank cell in the column in Problem 4, explain why it is blank. Comment on whether there are any figures with the same shape (not triangles or quadrilaterals).C. In some cells, there are two or more figures. Do these figures have anything in common other than the price? Discussion.D. You may have heard that “squares are squares, but squares don’t have to be squares.” Does the division of the four parts by the scale shed any light on this statement? Can you use symmetric division to find more expressions like this?74 Section 5 Symmetry Geometry Labs © 1999 Henri Picciotto, www.MathEducationPage.org LAB 5.3 Name(s) One Mirror Object: mirror, templateUsing a template your stitch on the piece that is not lined up. paper, draw a triangle. Then put a mirror on it so that it makes a triangle or four. For example, starting with a right triangle ABC it is possible to make a different triangle, kite, or rhombus. Check the figures you have made in this way in the table below, and draw a mirror line on the triangle as a record of your progress. you did. Also indicate the side you are looking at with an arrow. On the table, mark impossible with X. Figures made Using mirrors on Triangles EQ AI RI OI AS RS HE OSEquilateral (EQ)Acute isosceles (AI) Right isosceles (RI) Obtuse isosceles (OI) Acute scalene (AS) Right sikene (RS) Half-equal (HE) Scalene scalene (OS) Average Quadrilaterals EQ AI RI OI AS RS HE OSGeneralKiteGeneral trapezoidIsosceles trapezoidParallelogramRhombusRectangleSquareGeometry Labs Unit 5 Symmetry 75 © 199999 continued) DiscussionA. What types of polygons other than triangles and quadrilaterals can be made with a triangle and a mirror?B. What figures on the table cannot be made with a triangle and a mirror? Why? C. What do all the possible numbers do? D. What linear regression calculations cannot be performed? Why? Yes. What figures can be made from each triangle? F. What is a good formula for finding a way to create a given shape with a triangle and a mirror?G. How to put a mirror on the real triangle so that the figure becomes a triangle? Why? H. It is never possible to make an acute isosceles triangle from an obtuse isosceles triangle. However, sometimes it is possible to make an obtuse isosceles triangle from an obtuse triangle. Why?76 Section 5 Symmetry Geometry Labs © 1999 Henri Picciotto, www.MathEducationPage.org LAB 5.4 Name(s) Two Mirror Objects: Mirror Mirrors those who are bound by them. sticky tape, pattern blocks, template In this lab, you will investigate what happens to the reflection of the pattern blocks as you change the angle between the mirrors. 1. Using your mirror and one blue marker block, make the shapes below. To record your work, draw the position of the mirror on the figure.2. For each example in Problem 1, what is the angle between the mirrors? Explain. 3. What is the next figure in the series? Do