The drawing below shows a cube made of two identical pieces. Draw one of these pieces in 3 dimensions.
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To visualize 3D objects in LaTeX, use the `tikz` package for simple geometric shapes. For more complex 3D modeling, you can use specialized packages like `3dplot`. When drawing shapes like this L-shaped block, focus on one face at a time, and use perspective to connect them properly.
Step 1: Understand the Cube We are given a visual of a $3 \times 3 \times 3$ cube (based on the way it's subdivided), made from two identical interlocking shapes. Each piece must therefore occupy half of the total volume: A $3 \times 3 \times 3$ cube has 27 unit cubes. So, each identical piece must consist of 13.5 unit cubes, which is impossible if we're only using whole units. This means the cube is not made of a $3 \times 3 \times 3$ arrangement of unit cubes. So we need to reevaluate our assumption. Looking at the image carefully, we notice: - The cube is visually divided into two interlocking pieces. - The pieces appear to be mirror images or rotations of one another. - The cuts suggest 3D L-shapes or step-like extrusions. Let’s interpret the structure from the visible sides. Step 2: Analyze the Views There are 3 visible faces in the drawing: front, top, and right. Top View - There’s a square "notch" at the front-left. - This implies that a part of the block is missing from the top surface. Front View - Shows a tall vertical rectangle on the left and a thinner one next to it. - Suggests different height levels on the front of the block. Right View - Shows a vertical rectangular indentation, just like the front. - Suggests symmetry. Step 3: Visualize the 3D Piece Let’s try to reconstruct one of the pieces step by step. We assume: - The full cube is of size $2 \times 2 \times 2$ units (or 8 unit cubes). - This matches the general layout and allows each piece to have 4 unit cubes. Let’s label the unit cubes in a $2 \times 2 \times 2$ space: Layer 1 (bottom):[A1][A2]
[B1][B2]
Layer 2 (top):[C1][C2]
[D1][D2]
3D Breakdown of One Piece From the image and symmetry, one piece can be constructed from 4 unit cubes arranged like an inverted L-shape in 3D: - Bottom layer: 2 unit cubes in a straight line (forming a base). - Middle cube: One cube on top of one end of the base. - Top cube: One cube offset, forming a step-like shape. So, imagine this 3D L-shape where the pieces step upward and sideways. Alternatively, you can think of it as: - A vertical bar of 2 cubes. - At the base of the vertical bar, there is one cube extended horizontally. - On the top of the vertical bar, another cube is extended perpendicularly to the first. This results in a Z-shaped piece in 3D. Step 4: Final 3D Shape Description Let’s describe one piece in coordinate form for clarity: In a $3 \times 3 \times 3$ grid, one piece occupies the following unit cubes (each coordinate is of the form (x, y, z)):(0, 0, 0)
(1, 0, 0)
(1, 1, 0)
(1, 1, 1)
Here’s how this looks: - From (0,0,0) → (1,0,0): horizontal base - (1,0,0) → (1,1,0): vertical rise - (1,1,0) → (1,1,1): upward step This gives an interlocking L-shaped or stair-step shape that fits together with its mirror to form the cube. Final Answer: 3D Drawing Description You are expected to draw the piece in 3D. Here’s how you can do that: - Start with a cube at the bottom-left (base cube). - Add one cube to the right of it. - Add one cube on top of the right cube. - Add one cube on top of the last one, but shifted to the back. This forms an interlocking L or stair-like shape, which, when mirrored, fits perfectly with another identical piece to form a cube. Conclusion - The full cube is made of two identical interlocking 3D L-shaped pieces. - Each piece is a 4-cube structure arranged in a 3D step pattern. - Drawing one piece in 3D involves visualizing the step structure from the front to the back and from bottom to top.
