Boxes and Sketches - Perspective and Floor Maps

Difference between 2D and 3D: A 2-dimensional (2D) shape like a square or rectangle is flat and has only length and width. A 3-dimensional (3D) object, such as a box or a dice, has length, width, and depth (height). In sketches, we use 'deep drawings' to show the 3D nature of objects on a 2D piece of paper.

Nets of a Cube: A net is a flat 2D pattern that can be folded to form a 3D solid. For a cube, a net must consist of exactly $6$ squares. These squares must be arranged such that when folded, they do not overlap and they cover all sides of the cube. Visualizing a net involves mentally folding the squares to see if they meet at the edges to form a closed box.

Floor Maps: A floor map is a drawing that shows the layout of a space, like a room or a house, as seen from directly above. It displays the positions of walls, windows, and doors. On a floor map, windows are often represented by small gaps or lines on the perimeter walls, and doors are shown as openings with a small arc indicating the direction they swing.

Deep Drawings (Perspective): Unlike a floor map, a deep drawing shows a house or object with perspective, meaning it shows how it looks from the front and side to indicate its height and depth. A deep drawing helps us understand the volume of a structure, whereas a floor map only shows the boundary and internal divisions.

Connecting Maps to Drawings: To match a floor map to a deep drawing, you must check the relative positions of features. For example, if a floor map shows a window on the left wall and a door on the right wall, the corresponding deep drawing must show these features in the same relative positions when viewed from the front.

Faces, Edges, and Vertices: Every 3D box has specific properties. A face is a flat surface (a cube has $6$ faces), an edge is where two faces meet (a cube has $12$ edges), and a vertex (corner) is where edges meet (a cube has $8$ vertices). In a sketch, some edges and faces might be hidden from view depending on the perspective.

Perspective of Different Shapes: Objects look different depending on where you stand. A bridge viewed from the top might look like a simple long rectangle, while from the side, it shows its height, arches, and pillars. This change in appearance based on the viewpoint is a key part of making accurate sketches.