In addition to the 5 ways shown in Fig. 4.8, are there any additional ways of gluing four cubes together along faces? Can you visualise and draw these as well?

Question

Homi · Accepted Answer

Understand the Question

Four cubes connected face-to-face in a single plane form shapes known as tetrominoes.
If we only allow rotations in a plane (treating mirror reflections as distinct shapes), there are $7$ distinct one-sided tetrominoes.
Since Fig. 4.8 shows $5$ free tetrominoes (I, O, L, S, and T), the additional ways are formed by the reflections of the chiral shapes (L and S), giving the J-tetromino and Z-tetromino.

Step 1 · Identify the Given 5 Shapes

The $5$ planar arrangements (free tetrominoes) shown in Fig. 4.8 are:

I-tetromino: A straight line of $4$ cubes.
O-tetromino: A $2 \times 2$ square of $4$ cubes.
L-tetromino: An L-shaped arrangement.
S-tetromino: A zig-zag/S-shaped arrangement.
T-tetromino: A T-shaped arrangement.

Step 2 · Find and Visualise the Additional Reflected Shapes

When shapes that are mirror images (reflections) and cannot be matched purely by 2D rotation are considered distinct, there are $2$ additional arrangements:

J-tetromino: The reflection of the L-tetromino.
Z-tetromino: The reflection of the S-tetromino.

Answer

Yes, there are $2$ additional ways (considering reflections as distinct):

J-tetromino (reflection of the L-shape)
Z-tetromino (reflection of the S-shape)

Common Mistakes

Overlooking Reflections (Chirality): Assuming all rotations cover all orientations. In a 2D plane, the L-shape and S-shape cannot be rotated to form the J-shape and Z-shape without flipping them over (reflection).
Ignoring 3D vs. 2D Constraints: Conflating 2D planar tetrominoes with 3D non-planar tetracubes (which introduce additional non-flat arrangements in space).

Question 19

More questions in FIO