Boxes and Sketches - Drawing 2D representations of 3D objects

Difference between 2D and 3D: 2D shapes are flat drawings like squares or rectangles with only $length$ and $width$. 3D objects, like boxes or dice, have $length$, $width$, and $height$ (depth), allowing them to occupy space.

Nets of 3D Shapes: A net is a 2D pattern that can be folded along its edges to form a 3D object. For a cube, a net must consist of exactly $6$ squares. A common visual for a cube net is a 'cross' shape where four squares form a vertical column and two squares are attached to the sides of the second square.

Floor Plans: A floor plan is a 2D top-view drawing of a building or room. It shows the layout of walls, windows, and doors from above but does not show the height of the structure. It is like looking at a house through a camera positioned directly overhead.

Deep Drawings: Unlike a floor plan, a deep drawing is a 3D representation of an object on a 2D surface. It shows the front, side, and top parts of an object to give a realistic sense of its shape and depth. For a house, a deep drawing shows the roof, the windows on the side, and the front door simultaneously.

Visualizing Cubes: To draw a deep drawing of a cube, we start by drawing two overlapping squares of the same size. By connecting the four corresponding corners of these squares with diagonal lines, we create a 3D visual effect.

Mapping Nets to Boxes: Not every arrangement of $6$ squares can fold into a cube. For example, $6$ squares arranged in a single straight line cannot form a cube because the ends would overlap and leave the top/bottom open. A valid net must have faces that fold to meet at right angles without overlapping.

Counting Cubes in Sketches: When looking at a 3D sketch of stacked boxes, remember to count the hidden cubes. If a cube is visible on the second level, there must be another cube directly underneath it to support it, even if that bottom cube is not visible in the drawing.