Spatial Analysis Skills

Solve spatial problems by checking independent constraints

You have learned the PAT task types and basic transformations. Now combine them: derive a painted-face tally, reject an impossible projection, and explain why a tempting answer fails. Keep x = width, y = depth, z = height for the solid-object examples.

1. Count cubes with a height map

A height map records how many cubes stand in each vertical column. For the practice structure below, every occupied column starts at the table and has no gaps. Each bracket is one column position, not a perspective drawing.

Top-down height map:

             x=1  x=2
back, y=2    [1]  [1]
front,y=1    [2]  [1]

The map specifies five cubes: four on the bottom, one on top of the front-left cube. Call the bottom cubes A (front-left), B (front-right), C (back-left), D (back-right), and the upper cube E.

For each cube, begin with six faces. Subtract a face for each touching neighbor and subtract the bottom if it rests on the table. This example has no unsupported cubes or sealed cavities.

The painted-face tally is one cube with 2, three with 3, and one with 5. Total = 1 + 3 + 1 = 5 cubes, matching 2 + 1 + 1 + 1 in the height map. Total painted area is 2 + 3 + 3 + 3 + 5 = 16 unit faces; this checks the tally but is a different question from “how many cubes have three faces painted?”

In a PAT picture, infer only hidden cubes required for support. Do not fill space merely because a visible edge continues across it. The figure is cemented together; avoid claiming that every overhang is physically impossible in all real constructions. Use the ADA's construction and painting conventions.

Filled-block boundary check

A filled 4 × 3 × 3 block rests on the table; its top and four side surfaces are painted. How many cubes have exactly two painted faces?

The full tally is 4 zero-face, 14 one-face, 14 two-face, and 4 three-face cubes: 36 total. Painting all six outside surfaces instead would give 16 two-face cubes. The paint rule changes the answer even when the solid is identical.

2. Derive the missing view from what is supplied

PAT View Recognition supplies two orthographic views and asks for the third. Start with the available views; the top view may itself be missing. Check shared widths, depths, heights, step locations, and hidden edges against each candidate.

Worked practice model: a solid is built from three adjacent unit-width, unit-depth columns of heights 1, 2, 3, from left to right. There are no other columns. Its front elevation is a staircase and its end outline is a 1 × 3 rectangle. Choose the top view:

• A. One row of three unit squares, with two internal boundaries where the height changes.
• B. A 2 × 2 square.
• C. A triangular outline.
• D. One square of side 3.

Answer: A. Looking down collapses height but retains the three occupied positions and the edges at the steps. The top outer outline is rectangular, even though the front outline is stepped.

Top view of this specified solid:

      width = 3
    ┌───┬───┬───┐
    │   │   │   │  depth = 1
    └───┴───┴───┘

A view does not preserve every true edge length: an edge parallel to your viewing direction collapses to a point, and an oblique edge can be foreshortened. Compare aligned dimensions that should agree, rather than forcing every 2:1 edge ratio to appear unchanged in every view. A bottom view is not simply a copy of the top view's visible edges.

3. Define an axis before rotating

Only points on the rotation axis stay fixed. Corners and face centers elsewhere move along circular paths. “Rotate 90°” is incomplete unless the axis and direction are understood.

Take a cube with front F, back B, left L, right R, top U, bottom D. Rotate the whole cube 90° clockwise as seen from above, about its vertical z axis. Then:

The top face remains on top, though a mark on it rotates. After a 180° turn about this same axis, F is at the back, while U is still on top. “Every face becomes the opposite face after 180°” would be false.

Try it: follow R through two successive clockwise quarter-turns viewed from above: R → front → left. A counterclockwise turn would follow the opposite cycle.

4. Match an aperture using a specified solid

Imagine a rigid prism extruded along the y (depth) axis with constant L-shaped cross-section in the x–z plane. Its cross-section is the union of two rectangles: 0 ≤ x ≤ 1, 0 ≤ z ≤ 3 and 0 ≤ x ≤ 2, 0 ≤ z ≤ 1. It therefore has total width 2, height 3, and arms one unit thick. Orient its extrusion direction straight through the wall.

Cross-section; each # is one unit square:

       #.
       #.
       ##

An aperture with that exact outline and size matches the stated orientation. A similar L with both arms only half a unit thick is too small. A filled 2 × 3 rectangle may physically admit the object, but it does not match the required external outline for this PAT-style exercise. An internal decorative line would not change the aperture boundary.

In a full PAT item, consider other orientations before insertion; the picture does not lock the object in a “natural orientation.” Once passage begins, do not twist it. Match the whole transverse outline, not just a convenient notch.

5. Detect the tempting folding error

On a unit square, fold left onto right at x = 0.5 and then bottom onto top at y = 0.5. A small punch at (0.55, 0.55) passes through the four layers and does not touch a crease.

Undo the horizontal fold to add (0.55, 0.45), then the vertical fold to add (0.45, 0.55) and (0.45, 0.45). The four holes lie close to the center, one in each quadrant. A pattern near the four outside corners has the right count and the wrong positions.

Partial-fold counterexample: fold each outer third of a unit-width sheet onto its middle third. The two folds make three overlapping layers there, not four. An interior punch through them produces three holes. This is why “number of folds” cannot replace “number of layers at the punch.”

For a net, similarly avoid a count-only shortcut. In the cross net from the previous lesson, F–R–B and L–F–U each span two hinges. Only the first path is straight: F/B are opposite, whereas L/U are adjacent. Fold directions carry information that distance alone loses.

6. Order angles with the same reference

For labeled practice angles 1 = 30°, 2 = 180°, 3 = 60°, 4 = 90°, the increasing order is 1, 3, 4, 2. The first and third are acute; the fourth is right; the second is straight. The categories constrain the order, but the acute pair still needs comparison.

An angle of 45° is halfway between 0° and 90° in magnitude. It is not a universal judgment about whether a drawing looks “steep”: rotating the whole drawing changes its slope on the screen without changing its opening. For actual visual practice, compare the two rays at their vertex and follow the requested ascending order.

7. Review errors by their cause

After a short timed set, classify each miss: wrong task rule, missing information, incorrect transformation, missed face contact, or time pressure. Reconstruct the correct solution before trying another item. If using a physical object at home, make a prediction first and inspect the outcome second; otherwise you may only be watching a demonstration.

Use 40 seconds per PAT question as an overall average, with flexible allocation between tasks. Record which methods cost time without improving accuracy. Make a provisional choice before flagging; answer all items before the section ends. The current ADA exam page links the official examples for visual practice.