Question

1. use copies of the rectangle to show how a rectangle could:

a. tile the plane.
b. not tile the plane.

Explanation:

Part (a): Tile the Plane

Step 1: Understand Tiling

Tiling the plane means covering a flat surface with copies of the rectangle without gaps or overlaps. For a rectangle, we can use its sides (length \( l \) and width \( w \)) to repeat the shape.

Step 2: Arrange Rectangles

Take the given rectangle (let's assume its dimensions are, e.g., 3 units by 2 units from the grid). Place one rectangle, then place another adjacent to it along the length: the length of the first rectangle (3 units) aligns with the length of the second. Then, stack rows of these rectangles. Since rectangles have right angles, adjacent rectangles will fit together (the \( 90^\circ \) angles will combine to \( 180^\circ \) or \( 360^\circ \) as needed, and the side lengths match when repeated). For example, if the rectangle is \( 3 \times 2 \), we can place multiple \( 3 \times 2 \) rectangles side - by - side (along the 3 - unit side) and row - by - row (along the 2 - unit side) to cover the grid (plane) without gaps or overlaps.

Part (b): Not Tile the Plane

Step 1: Disrupt Tiling Conditions

To not tile the plane, we need to arrange the rectangles such that there are gaps or overlaps. One way is to rotate the rectangles in a way that their sides don't align properly. For example, if the original rectangle has length \( l \) and width \( w \), rotate some rectangles by \( 90^\circ \) and place them in a non - repeating, non - aligned pattern. Suppose the rectangle is \( 3 \times 2 \). If we place one rectangle normally (length 3 horizontal, width 2 vertical), then place a rotated rectangle (length 2 horizontal, width 3 vertical) next to it in a way that the sides don't match up. The rotated rectangle's length (2) and the original's length (3) don't have a common multiple in the way that allows seamless tiling, leading to gaps between the rectangles or overlaps if forced. Another way is to place rectangles with inconsistent spacing: leave spaces between them or overlap parts of the rectangles, which violates the tiling rules (no gaps, no overlaps).

Answer:

s (Visual Explanations as Described):

Part (a)

Draw multiple copies of the given rectangle, placing them adjacent to each other (horizontally and vertically) such that their sides align perfectly. For example, if the rectangle has a length of 3 grid squares and a width of 2 grid squares, place one rectangle, then place another to the right of it (sharing the 2 - unit width side, so the 3 - unit length sides are adjacent), and continue this horizontally. Then, place rows of these horizontal strips vertically, with each row's rectangles aligned with the row above/below. This will cover the grid (plane) without gaps or overlaps.

Part (b)

Draw copies of the rectangle with some rotated (e.g., 90 - degree rotation) and placed in a way that their sides do not align. For example, place one rectangle normally, then place a rotated rectangle next to it so that the length of the rotated rectangle (which was the width of the original) and the length of the original rectangle do not match up, creating a gap. Or, place rectangles with intentional spaces between them or overlapping parts, so that the plane is not covered uniformly without gaps/overlaps.