9.4 Spatial Rotations (2D and 3D)

Turning figures is not the same as flipping them

Spatial rotation items ask what a figure looks like after a turn, or which cube matches a net, or which face sits opposite another. They still sit in Core Abilities — analytical aptitude, not in Mechanical Reasoning. You are not asked which gear would reverse direction. You are asked whether two pictures are the same object after a turn in the plane or in space.

QFD does not publish a split between 2D and 3D items, a cube-net count, or a time-per-item figure. This section teaches the two distinctions that decide most independent practice items: rotation versus reflection in 2D, and adjacent versus opposite on a cube.

OnVUE still forbids paper. You cannot fold a printed net. Fold it mentally and store a verbal landmark: the short arm of F points right; after 90° clockwise it points down. Follow the candidate rules Pearson shows on the sitting.

2D rotation keeps handedness

A rotation turns a figure around a point in the page. Every point travels on a circle. Handedness stays the same: if the original R has a closed loop on the left of a vertical stem and a diagonal leg kicking right, a rotated R still has that loop on the same side of the stem relative to the leg. You can turn a piece of paper on the desk (in real life) and the R remains an R.

A reflection (mirror, flip) swaps left and right. The loop of R jumps to the other side of the stem. No amount of turning the paper on the desk produces that mirror-R. You would need to flip the paper onto its back. A mirror is not a rotation.

Worked example — the letter F. Describe an outline F as a vertical stem, a short top arm pointing right, and a shorter middle arm also pointing right. After a 90° clockwise rotation, the stem lies horizontal, the former top arm now points down, and the middle arm also points down, still on the same side of the stem. After 180°, the stem is vertical again, both arms point left, and the figure looks like an upside-down, backward F that is still the same handedness. A mirror F through a vertical axis has both arms pointing left while the stem is still upright and the top arm is still at the top — that is a flip, not a 180° turn. The 180° turn also puts the arms at the bottom relative to the new stem orientation if you track the serif that used to be the top. Track one landmark at a time: where did the top arm go, and is it still on the original side of the stem?

Worked example — shaded half-disk. A circle is split by a diameter. The right half is filled, the left half is empty, and a small notch sits on the top of the circle. Rotate 90° clockwise: the filled half moves to the bottom, and the notch moves to the right. A mirror through a vertical axis would keep the notch at the top and move the filled half to the left. If an option keeps the notch at the top and fills the left, it is a reflection, not a 90° rotation. Candidates miss this when they only track fill and forget the notch, or only track the notch and forget fill.

Worked example — L made of squares. The home L has a vertical stack of two squares and a foot of one square sticking right from the bottom. Rotate 90° clockwise: the long arm lies horizontal to the right, and the foot now sticks down from the right end. The mirror of the home L has the foot sticking left. That mirrored L is the same isolate you met in Section 9.2. On a same or rotated? item, the mirrored L is not a rotation of the home L.

Symmetric figures are the exception that proves the rule. A plus sign or a plain square with no extra marks looks the same after many rotations and after some reflections. Test writers who want a rotation-versus-mirror item add an asymmetric landmark: a thick border on one side, a missing corner, a single dot. Hunt that landmark first. If you cannot find an asymmetry, every option may be a rotation, and the item is probably testing something else (count, not pose).

3D cubes: adjacent, opposite, and corners

A cube has six faces. Each face has one opposite face that never shares an edge. The other four faces are adjacent to it.

A corner view shows three faces that meet at one vertex. Those three faces are all adjacent to one another. A pair of opposite faces can never appear together in a corner view. If a question shows a cube with A, B, and C meeting at a corner, then A is not opposite B, A is not opposite C, and B is not opposite C. The opposites of A, B, and C are the three faces you cannot see.

Worked example — opposite pairs from a corner. Suppose one picture shows faces A, B, and E meeting at a corner. Then the opposite of A is not B and not E. A later net or a later view that claims A opposite B is inconsistent with that corner. If the stem tells you A is opposite C, B is opposite D, and E is opposite F, then a legal corner is any trio that takes one from each pair, for example A, B, E or C, D, F. An illegal corner is A, C, and E, because A and C are opposites and cannot meet.

Worked example — which face is on the bottom? A cube sits with A on top and B facing you. If A is opposite C, then C is on the bottom. The four side faces are B and D (if those are a pair, they cannot both be sides in a way that puts them adjacent — if B is opposite D, then D is the back, not a left or right). Left and right would then be E and F. You do not need to invent extra letters. Track: top's opposite is bottom; front's opposite is back.

Nets: folding without paper

A net is a flat pattern of six squares that can fold into a cube. Not every hexomino is a valid net; for study, assume the item's net is valid unless the question asks which net cannot fold.

Two facts do most of the work:

1. Squares that share an edge on the net become adjacent faces on the cube.
2. In a straight strip of four squares, the first is opposite the third, and the second is opposite the fourth, once the strip is wrapped around the cube's belt.

Worked T-net (teaching labels). Build a net with a four-square belt and two flaps:

• A front square.
• Left on the front's left edge, right on the front's right edge, and back on the right's outer edge, making the straight four-strip left–front–right–back.
• Up on the front's top edge.
• Down on the front's bottom edge.

Apply the two facts. Shared edges are adjacent, so front is adjacent to left, right, up, and down — never opposite any of those four. The four-strip rule then names the opposites: position 1 opposite position 3, and position 2 opposite position 4. That gives left opposite right and front opposite back. Up and down attach to opposite edges of the same square (front), so after folding they land on opposite faces: up opposite down. Those three pairs fill the cube.

If a question asks which face is opposite front, the four-strip answer is back. An option that says up is opposite front has confused a flap that shares an edge with front for an opposite. Shared edge means adjacent, never opposite.

Firefighter flavour, still spatial rather than mechanical: folding a cardboard stores carton so that the label on the net's up square ends on top of the packed cube is the same opposite-and-adjacent test. You are not asked about the carton's strength.

A no-paper method

1. Decide 2D or 3D. If the figure is flat with no folds, use handedness. If six faces or a net appear, use opposite versus adjacent.
2. Pick one asymmetric landmark and move only that landmark through the stated angle.
3. Ask: did left and right swap while the top stayed the top? If yes, it is a mirror, not a rotation.
4. On cubes, write three pairs (even mentally): top/bottom, front/back, left/right. A new view must respect those pairs.
5. On nets, mark shared edges as adjacent and apply the four-strip opposite rule before you fold anything fancy.
6. Reject physics stories. Which face is opposite is not which face would bear the load.

Traps on spatial rotation

• Calling a mirror a 180° rotation because both can look backward.
• Tracking fill and ignoring a notch, or tracking a notch and ignoring fill.
• Putting two opposite faces on the same corner view.
• Treating a net neighbour as an opposite.
• Folding a flap the wrong way so up lands on front (they share an edge; they stay adjacent).
• Importing Mechanical Reasoning (the cube would roll downhill that way).

Pearson's Sample Demonstration is for OnVUE controls. It is not a published cube syllabus. Independent practice in this chapter is for the analytical-aptitude element the 2024 pack names. Do not study toward an invented number of rotation items, and do not treat OpenExamPrep figures as QFD's own.

On that net the belt is the four-strip Left–Front–Right–Back, so Left is opposite Right and Front is opposite Back. Up and Down attach to opposite edges of Front, so Up is opposite Down. Front shares an edge with Up, so Up is adjacent to Front, never opposite Front. If a sitting shows a cube with Front, Right, and Up meeting at a corner, that trio is legal because none of those three pairs is an opposite pair. A corner that showed Front and Back together would be illegal.