Question

A coloured sheet is folded into an octahedron as shown on the left. Which of the options show(s) the CORRECT rotated views of the octahedron?

• A. Fig A
• B. Fig B
• C. Fig C
• D. Fig D

Hint:

For any regular octahedron net in a straight strip of 8 triangles, opposite faces are always separated by exactly 3 intermediate faces (i.e., \(F_i\) and \(F_{i+4}\)).
Since opposite faces cannot share a vertex, you can instantly eliminate any options showing opposite color pairs meeting.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question requires us to identify the correct rotated 3D views of an octahedron after folding a 2D net of eight colored triangles.
The net consists of eight alternating triangles with a specific sequence of colors: Green, Yellow, Green, Orange, Blue, Orange, Blue, and Yellow.

Step 2: Key Formula or Approach:
To solve folding problems of regular polyhedra, we trace the face adjacency and determine opposite pairs.
In a standard regular octahedron formed by a straight strip of eight alternating triangles:

• Face \(i\) is always directly opposite to face \(i+4 \pmod 8\).

• Opposite faces in an octahedron do not share any vertex or edge.

• The colors of the faces from 1 to 8 are:
\(F_1 = \text{Green}\), \(F_2 = \text{Yellow}\), \(F_3 = \text{Green}\), \(F_4 = \text{Orange}\),
\(F_5 = \text{Blue}\), \(F_6 = \text{Orange}\), \(F_7 = \text{Blue}\), \(F_8 = \text{Yellow}\).


Step 3: Detailed Explanation:

• Let us identify the opposite pairs using our rule:
- \(F_1\) (Green) is opposite to \(F_5\) (Blue).
- \(F_2\) (Yellow) is opposite to \(F_6\) (Orange).
- \(F_3\) (Green) is opposite to \(F_7\) (Blue).
- \(F_4\) (Orange) is opposite to \(F_8\) (Yellow).

• This means a Green face can never meet a Blue face at a shared edge, and a Yellow face can never meet an Orange face at a shared edge.

• Let us analyze the options:
-

Option B: Shows a view from a vertex where Green, Yellow, Yellow, and Blue meet.
The two Yellow faces (\(F_2\) and \(F_8\)) are diagonally opposite, and the Green (\(F_3\)) and Blue (\(F_5\)) are diagonally opposite.
Since these diagonal pairs are not opposite faces on the octahedron, they can meet at a common vertex. This is a geometrically valid configuration.
-

Option C: Shows a view where Green, Orange, Orange, and Blue meet.
By the symmetry of the net (swapping Green with Blue and Yellow with Orange), this is the exact symmetric counterpart to Option B.
Therefore, Option C is also a correct rotated view.


Step 4: Final Answer:
Options (B) and (C) represent correct rotated views of the folded octahedron.