Isometry on Asymmetrical Shapes
Investigate the effect of isometries on an asymmetric shape
- Perform the four isometric transformations on the shape below: translation, rotation, reflection, and glide reflection.
- Record what characteristic changes and what stays the same (side lengths, angles, orientation, shape etc).
The Algebra of Movement in Isometric Transformation
Goal: In transformations, there are some movements that have points that do not move. Can you identify them?
Invariant points: a fixed point that do not change its position/ stay still when a transformation is performed.
- Create a shape on the grid below. Perform a reflection.
- Does any point on the shape stay in the same place?
- Drag the shape so it touches/crosses the line of reflection. Does any point stay in the same place?
- Now, perform a rotation. Move the center point of rotation. What happens to the center point itself?
- Lastly, perform a translation. Do any point stay still?
- Can you conclude which isometric transformation has invariant point(s)?
The 'Home' Move (Inverse):
- Create a shape on the grid.
- Perform translation, . Now, perform another transformation so that it maps the image back onto the original object (perform the 'home' move). What is a single transformation that moves the image exactly back onto its object position?
- Repeat with a clockwise rotation. What is a single transformation that moves the image exactly back onto its object position?
The 'Double' Move (Product):
- Create a shape on the grid.
- Perform a reflection over a line . Reflect that new image over another vertical line .
- Observe the original object and the final image. What single transformation could have skipped the middle step?
- Repeat with a rotation followed by another rotation. What single transformation could have skipped the middle step?
- Create a shape on the grid.
- Perform a clockwise rotation.
- Observe what changed.
Abstraction
Identity transformation: is a mapping where every point stays in its original position. It is the "zero" of transformations.
For any transformation , the composition and where the identity results in no change.
The Inverse is the transformation that maps the image back to the original object.
Every transformation has 'reverse' move, where the inverse of a transformation is , such that the composition or
The Product is a composition of two or more transformations in a sequence to create a new isometry.
For transformation followed by , their product is also an isometry.
Note:
- Rotations have one invariant point; reflections have a line of invariant points; translations have none.
- A rotation or a double reflection over the same line results in the identity.
Direct vs. Opposite Isometry
Goal: Use vertex labeling to distinguish between "sliding/turning" and "flipping."
- Create a shape. Label all vertices in clockwise order (A, B, C, ...).
- Perform each isometric transformation on the shape. Record the orientation of the image. Which transformation results in the order remaining clockwise, and which is counter-clockwise? Did it 'flip'?