📐 Figure Series – Advanced Study Notes
For the “Reasoning Ability” section of the HTET (Higher Technical Entrance Test)
Prepared by a MIT‑level teacher: from first principles to cutting‑edge theory.
1. Introduction
| Term | Definition | Example |
|---|---|---|
| Figure Series | A finite sequence of two‑dimensional diagrams (shapes, polygons, symbols) that follow a hidden rule. The test taker must infer the rule and predict the next element. | ① ▲ ② △ ③ ◊ ④ □ ⑤ … |
| Reasoning Ability | Aptitude to recognize patterns, relationships, and logical structures. Figure series test this by stressing visual and spatial reasoning. |
Why do we study it?
In logic and CS, pattern recognition is an instance of inductive inference. Understanding figure series trains the brain to abstract rules, generalize, and verify hypotheses—a skill transferable to algorithm design, geometry, and AI.
2. Formal Foundations
2.1. Basic Notation
Let
- (S = (F_1, F_2, \dots, F_n)) be a sequence of figures.
- Each figure (F_i) is a finite set of points in (\mathbb{R}^2).
- The rule (T) is a function (T: \mathbb{R}^2 \to \mathbb{R}^2) (or a family of such functions) such that
[ F_{i+1} = T(F_i), \quad i = 1,2,\dots,n-1. ]
2.2. Transformations as Group Actions
A transformation (T) is an isometry (distance‑preserving) or a similarity (distance‑preserving up to scaling). The set of all such transformations forms a group under composition.
- Euclidean group (E(2)): translations, rotations, reflections.
- Similarity group (\mathrm{Sim}(2)): Euclidean group plus uniform scaling.
Key Observation
Most figure‑series rules can be expressed as elements of a (sub)group of (E(2)) or (\mathrm{Sim}(2)).
For example, a sequence that rotates the figure by (90^\circ) each step corresponds to the cyclic subgroup (\langle R_{90^\circ}\rangle).
3. Classification of Elementary Patterns
| Category | Typical Transformation | Example | Remarks |
|---|---|---|---|
| Rotational | (R_\theta) | ① ▲ → ② △ (rotate 60°) | Closed under composition; order matters. |
| Reflective | Reflection across a line | ① ⬛ → ② ⬜ (mirror) | Two reflections = rotation. |
| Translational | (T_{\vec{v}}) | ① ⬛ → ② (shift right) | Often combined with scaling. |
| Scaling | (S_k) | ① ⬛ → ② (double size) | Non‑invertible if (k=0). |
| Color Change | Attribute mapping | ① ⬛ (red) → ② (blue) | Requires separate attribute set. |
| Shape Substitution | Shape → other shape | ① ⬛ → ② △ | Often paired with symmetry operations. |
| Composite | Combination of above | ① ⬛ → ② (rotate + scale) | Common in exam questions. |
4. Algebraic Representation of Figures
4.1. Coordinates & Matrices
Represent a figure (F) as a finite set of column vectors ({ \mathbf{p}_1, \dots, \mathbf{p}_m }).
A linear transformation (T) is a matrix (A \in \mathbb{R}^{2 \times 2}) and a translation vector (\mathbf{t}). Then
[ T(\mathbf{p}) = A \mathbf{p} + \mathbf{t}. ]
Example: A rotation by (\theta) is [ A = \begin{bmatrix} \cos\theta & -\sin\theta\ \sin\theta & \cos\theta \end{bmatrix}. ]
4.2. Composition & Group Law
Given (T_1 = (A_1,\mathbf{t}_1))