Goa TET – Reasoning Ability
Figure Series – Advanced Study Notes (MIT‑Level)
Goal – Understand Figure Series from first principles, develop a rigorous analytical framework, and master the art of spotting the hidden rule.
Level – Undergraduate (MIT) – deep geometry, symmetry groups, and algorithmic pattern detection.
1. Introduction
A figure series is a sequence of two‑ or three‑dimensional shapes where each term is a transformation of the previous one. The task is to:
- Identify the transformation rule.
- Predict the next figure(s).
- (Optional) Explain why other plausible rules are not correct.
Unlike simple “count‑the‑lines” tricks, figure series require conceptual geometry and invariant reasoning.
2. First Principles: Geometry & Transformations
2.1 Euclidean Plane Geometry
- Points, lines, angles, distances.
- Basic shapes: triangles, quadrilaterals, circles.
- Congruence: two figures are congruent iff they can be superimposed by a sequence of rigid motions.
2.2 Transformations in the Plane
| Transformation | Symbol | Algebraic effect on coordinates ((x,y)) | Key property |
|---|---|---|---|
| Translation | (T_{\mathbf{v}}) | ((x,y)\to (x+v_x,,y+v_y)) | Vector addition |
| Rotation | (R_\theta) | ((x,y)\to (x\cos\theta - y\sin\theta,; x\sin\theta + y\cos\theta)) | Distance from origin preserved |
| Reflection | (S_\ell) | Reflect across line (\ell) | Mirror image |
| Dilation (Scaling) | (D_k) | ((x,y)\to (k x,,k y)) | Uniform scaling from origin |
| Glide Reflection | (G_{\mathbf{v},\ell}) | (S_\ell) followed by (T_{\mathbf{v}}) | Combines reflection + translation |
Composite transformations are formed by concatenation, e.g. (R_{90}\circ S_{y}). Every Euclidean isometry can be written as a product of at most three of these.
2.3 Group‑Theoretic View
Let (E(2)) be the Euclidean group of all isometries in (\mathbb{R}^2).
- Subgroups: rotations (SO(2)), reflections (O(2)), translations (\mathbb{R}^2).
- The action of (E(2)) on a set of figures (F) defines orbits (equivalence classes).
- The stabilizer of a figure (f\in F) is the set of transformations that leave (f) unchanged.
Key Insight – In a figure series, the transformation from term (n) to (n+1) is chosen from a finite subset of (E(2)). The rule is often a sequence of group elements: (g_1, g_2, g_3, \dots).
3. Pattern Classification
3.1 Transformation Types
| Category | Typical Features | Common Series |
|---|---|---|
| Pure translations | Shift by a fixed vector | “Shift right 2 units” |
| Rotations | Fixed angle, center may move | “Rotate 90° about the centre” |
| Reflections | Across a fixed line | “Mirror left‑right” |
| Dilation | Size changes by factor | “Double the size” |
| Affine combinations | Linear mapping + translation | “Stretch horizontally by 2” |
| Composite | Two or more of the above | “Rotate 90°, then reflect” |
3.2 Composite Transformations & Their Signatures
- Rotation + Reflection → Glide reflection or a rotation about a different centre.
- Scaling + Translation → Similarity transformation (maintains shape).
- Rotation + Scaling → Similarity with changing orientation.
Signature extraction: look for invariant properties:
- Side lengths → scaling invariant if ratio preserved.
- Angles → rotation or reflection preserve angles.
- Area → dilation changes it; reflection/rotation preserves.
3.3 Edge Cases
| Edge | Description | Why it’s tricky |
|---|---|---|
| Multiple plausible rules | Two different transformations explain the sequence | Must eliminate via “next term” or “further detail” |
| Hidden dimensions | 3D figure projected into 2D, giving illusion of shape change | Detect by checking orthogonality or hidden lines |
| Over‑specification | Figures look similar but have subtle differences (e.g., a small angle change) | Requires precise measurement or invariant calculation |
| **Non |