Gbk geometry
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GBK Geometry. Jordan Johnson. Today’s plan. Greeting Hand in Problem Write-up Lesson: Isometries, Congruence, and Symmetry Study Guide Clean-up. On parallelograms. Roger Shepard, 1981. Isometries / Rigid Motions.

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GBK Geometry

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Gbk geometry

GBK Geometry

Jordan Johnson


Today s plan

Today’s plan

  • Greeting

  • Hand in Problem Write-up

  • Lesson: Isometries, Congruence, and Symmetry

  • Study Guide

  • Clean-up


On parallelograms

On parallelograms


Roger shepard 1981

Roger Shepard, 1981


Isometries rigid motions

Isometries / Rigid Motions

  • A transformation is an isometry (or rigid motion) iff it preserves distances and angles.


Example isometry proof rotation preserves distance

Example Isometry Proof:Rotation Preserves Distance


Example isometry proof rotation preserves distance1

Example Isometry Proof:Rotation Preserves Distance


Example isometry proof rotation preserves distance2

Example Isometry Proof:Rotation Preserves Distance


Example isometry proof rotation preserves distance3

Example Isometry Proof:Rotation Preserves Distance

1

2

3


Definitions

Definitions

  • Transformations  definition of congruence:

    • Two figures are congruent iff there is an isometry in which one is the image of the other.


Glide reflections

Glide Reflections


Definition

Definition

  • A glide reflection is the composite of a translation and a reflection in a line parallel to the direction of the translation.

    • In other words: Reflect, then translate, or vice versa.


Reflection symmetry

(Reflection) symmetry


2 fold rotational symmetries

2-fold Rotational Symmetries


3 fold rotational symmetry

3-fold Rotational Symmetry

Flag of the Isle of Man


Translational symmetry

Translational Symmetry

Other neat examples: autologlyphs and ambigrams.


A tale of two fats

A Tale of Two Fats

Saturated

Unsaturated


Translational symmetry1

Translational Symmetry


That form blocks

…that form blocks…


Like lard

…like Lard.


Unsaturated fats

Unsaturated Fats


Summary

Summary

  • Each isometry has an associated symmetry:

    • Reflection symmetry with respect to a line:

      • Figure coincides with its reflection image through the line

    • Rotation symmetry:

      • Figure coincides with its image when rotated less than 360°

      • n-fold symmetry, if the smallest such rotation is 360°⁄n

    • Translation symmetry:

      • Figure coincides with its translation image


Homework

Homework

  • Problem for final

  • Study guide

  • Any make-up portfolio work

  • Recommended review for Unit 5:

    • Proof problems:

      • Two reflections across parallel lines= translation perpendicular to the lines of reflection.

      • Prove that the 3 transformations are isometries.

    • Asgs #37-38.


Clean up reminders

Clean-up / Reminders

  • Pick up all trash / items.

  • Push in chairs (at front and back tables).

  • See you tomorrow!


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