2.3 Mirror Images, Water Reflections & 3D Figure Nets
Key Takeaways
- A vertical mirror image reverses left and right coordinates while preserving top and bottom vertical orientations.
- A water reflection acts as a horizontal mirror image, inverting top and bottom positions while maintaining left and right alignment.
- In a standard 6-square cube net, faces separated by exactly one intervening square are opposite faces and can never be adjacent on the folded cube.
- There are exactly 11 distinct valid net patterns that can be folded into a 3D cube.
- When folding a 3D pyramid net, all triangular side faces must meet at a single apex point directly above the base polygon.
Mirror Images vs. Water Reflections
Visual reflection sub-tests evaluate a candidate's mental mapping across reflection axes. In PAF intelligence tests, two distinct types of reflection problems appear regularly: Vertical Mirror Images and Water Reflections (Horizontal Mirror Images).
1. Vertical Mirror Images (Left-Right Reversal)
A vertical mirror is placed to the left or right of a figure (or text string). The fundamental rule of vertical mirror reflections is:
Key Vertical Reflection Principles
- Proximity Rule: Features closest to the mirror line in the original object must remain closest to the mirror line in the reflected image.
- Lateral Inversion: Asymmetrical symbols flip horizontally (for example, 'E' becomes '3', 'P' becomes 'q'-like shape, 'R' reverses).
- Symmetrical Invariants: Symmetrical letters like 'A', 'H', 'I', 'M', 'O', 'T', 'U', 'V', 'W', 'X', 'Y' remain unchanged in vertical mirror reflections.
2. Water Reflections (Top-Bottom Inversion)
A water reflection assumes a horizontal mirror plane placed underneath the object (or above it). The fundamental rule of water reflections is:
Comparison Table: Vertical Mirror vs. Water Reflection
| Feature / Axis | Original Figure | Vertical Mirror (Side Axis) | Water Reflection (Bottom Axis) |
|---|---|---|---|
| Arrow Pointing Top-Right | Pointing Top-Right | Pointing Top-Left | Pointing Bottom-Right |
| Letter 'P' | P | q (Lateral flip) | b (Upside down) |
| Clock Time 3:00 | Hands at 12 & 3 | Hands at 12 & 9 (9:00) | Hands at 6 & 3 (8:30) |
| Shaded Top-Left Corner | Top-Left shaded | Top-Right shaded | Bottom-Left shaded |
Step-by-Step Reflection Elimination Strategy
- Identify the Reflection Axis: Check if the problem asks for a mirror image (vertical plane) or water image (horizontal plane).
- Select a High-Contrast Feature: Focus on a single distinctive element located at an extreme corner (for example, a shaded dot in the top-right).
- Apply the Axis Rule: For vertical mirror, move top-right feature to top-left. For water reflection, move top-right feature to bottom-right.
- Eliminate all options that fail this single feature placement.
3D Nets, Folding Rules & Spatial Assembly
A 3D Figure Net is a two-dimensional planar layout of polygons that can be folded along its edges to construct a three-dimensional solid (such as a cube, triangular prism, or pyramid). In PAF Airman tests, candidates must match a 2D net to its corresponding folded 3D object, or identify impossible folded configurations.
Cube Nets & Structural Rules
A cube has 6 square faces, 12 edges, and 8 vertices. There are exactly 11 valid nets that can fold into a closed 3D cube (for example, 1-4-1 nets, 2-3-1 nets, 3-3 nets, and the 2-2-2 stair net).
The Opposite Face Rule (Crucial Shortcut)
In any linear strip of squares within a cube net, faces that are separated by exactly one intervening square are OPPOSITE faces on the folded 3D cube.
Key Consequences of Opposite Faces
- Opposite faces can NEVER touch or share an edge in the folded 3D cube.
- In a 3D view showing 3 visible adjacent faces of a cube, you can NEVER see two opposite faces simultaneously.
- If a folded cube option shows two opposite faces touching at an edge or appearing together in the 3D isometric view, that option is GEOMETRICALLY IMPOSSIBLE and must be eliminated immediately.
3D Net Face Relationship Table
| Net Pattern Type | Example Layout Description | Opposite Face Pairs |
|---|---|---|
| 1-4-1 Net (Standard Cross) | Central strip of 4 squares with 1 top flap and 1 bottom flap | Squares 1 & 3 of strip; Squares 2 & 4 of strip; Top & Bottom flaps |
| 2-3-1 Net | Offset rows of 2, 3, and 1 squares | Alternate squares in long rows |
| 3-3 Net (Z-Shape) | Two parallel rows of 3 squares offset by 1 | Squares separated by 1 step across step bends |
| Triangular Pyramid (Tetrahedron) | Central triangle surrounded by 3 equilateral flap triangles | Base triangle is opposite the apex meeting point of flaps |
Corner & Edge Adjacency Tracking
When folding a net, edges that meet at a 90° internal corner in the flat net will join together to form a single 3D edge. Always verify that shaded patterns or arrows along these meeting edges align correctly in the folded figure.
What is the correct water reflection (horizontal mirror image at the bottom) of a clock showing the time 3:00 (hour hand at 3, minute hand at 12)?
In a standard 1-4-1 linear strip cube net with 4 aligned squares labeled 1, 2, 3, 4 from left to right, which pair of faces will be opposite each other when folded into a 3D cube?
If Face A and Face B are opposite faces on a 3D cube, which of the following statements must be true when viewing the folded cube in an isometric 3D perspective showing 3 visible faces?