3.2 Visual Analogies, Geometric Transformations & Proportion Mapping

3.2 Visual Analogies, Geometric Transformations & Proportion Mapping

Quick Method: A TR-YÖS Visual Analogy (Şekil İlişkileri / Analojiler) presents a proportional relationship $A : B :: C : ?$ (read as "Figure $A$ relates to Figure $B$ in the exact same way that Figure $C$ relates to the unknown target figure"). Rather than guessing vaguely, express the relationship as a composite operator $T = T_k \circ \dots \circ T_1$ satisfying $B = T(A)$, verify whether each sub-operation is a reflection, rotation, inside-outside inversion, shading negation, or quantitative scaling, and apply $T(C)$ step by step.

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Structure of TR-YÖS Visual Analogies ($A : B :: C : ?$)

On the Sayısal Yetenek Testi, visual analogy items test whether you can abstract a transformation rule independent of the specific shapes on which it acts. For example, if Figure $A$ is a triangle that turns upside down and swaps its inner dot color to become Figure $B$, and Figure $C$ is a pentagon, you cannot simply look for another triangle—you must apply the exact operator (vertical flip + dot color negation) to the pentagon $C$.

This guide groups candidate analogy operations into four useful families:

1. Isometric Planar Transformations: Reflections across an axis and rotations about a center.
2. Topological Layer Permutations: Inside-outside inversions (İç-Dış Yer Değiştirme) of nested shapes.
3. Color, Shading, and Hatching Logic: Binary complementation or directional hatching flips.
4. Quantitative Proportion Mapping: Scaling internal partitions or mapping polygon side counts to discrete element counts.

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1. Reflections & Rotations: Mastering Orientation and Chirality

The Four Fundamental Reflections

A planar reflection flips every point across a fixed mirror line (axis of symmetry):

• Horizontal Reflection (Dikey Eksene Göre Simetri / Left-Right Flip): The mirror is a vertical axis ($x = 0$), mapping $(x, y) \mapsto (-x, y)$. Features on the left move to the right; top and bottom coordinates remain unchanged.
• Vertical Reflection (Yatay Eksene Göre Simetri / Top-Bottom Flip): The mirror is a horizontal axis ($y = 0$), mapping $(x, y) \mapsto (x, -y)$. Features at the top move to the bottom; left and right coordinates remain unchanged.
• Main Diagonal Reflection ($y = x$): Swaps horizontal and vertical coordinates $(x, y) \mapsto (y, x)$, sending North $\leftrightarrow$ East and South $\leftrightarrow$ West.
• Anti-Diagonal Reflection ($y = -x$): Maps $(x, y) \mapsto (-y, -x)$, sending North $\leftrightarrow$ West and South $\leftrightarrow$ East.

Distinguishing a $180^\circ$ Point Rotation from a Single-Axis Reflection

One of the most common mistakes on TR-YÖS is confusing a vertical flip (or horizontal flip) with a $180^\circ$ rotation. For symmetric shapes like an isosceles triangle pointing North, both a vertical reflection and a $180^\circ$ rotation make the triangle point South. However, as soon as an asymmetric feature is attached—such as an L, F, P, R, or S-shaped hook, or a corner dot on the left base—the two operations yield completely different figures!

Mathematically, a $180^\circ$ rotation about the origin maps $(x, y) \mapsto (-x, -y)$, which is equivalent to performing both a horizontal reflection and a vertical reflection in succession:

$$R_{180^\circ} = \text{Flip}_{\text{horizontal}} \circ \text{Flip}_{\text{vertical}}$$

• Chirality (Handedness) Rule: Every single reflection reverses chirality (a clockwise-curving hook becomes a counterclockwise-curving hook; an F facing right becomes a backward F facing left). Every pure rotation ($90^\circ$ CW, $90^\circ$ CCW, or $180^\circ$) is a rigid motion in the 2D plane and preserves chirality (representing an even number of reflections).
• Practical Test on Asymmetric Hooks (L / F / S): Pick the spine of an F or L shape and walk from its base to its tip. If the short horizontal arm sticks out to your right as you walk from base to tip in Figure $A$, then after any pure rotation ($90^\circ$ or $180^\circ$), walking from base to tip will still have the short arm sticking out to your right. If the short arm sticks out to your left when walking from base to tip, the figure underwent an odd number of reflections!

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2. Inside-Outside Inversion (İç-Dış Yer Değiştirme)

• 2-Layer Swap: If Figure $A$ consists of an outer shape $S_{\text{out}}$ enclosing an inner shape $S_{\text{in}}$, then $B = T(A)$ expands $S_{\text{in}}$ to become the new outer boundary and shrinks $S_{\text{out}}$ to become the new inner core:

$$(S_{\text{out}},\; S_{\text{in}}) \longmapsto (S_{\text{in}},\; S_{\text{out}})$$

• 3-Layer Cyclic Permutation: When three concentric shapes are present—Outer ($O$), Middle ($M$), and Inner ($I$)—never assume a simple 2-swap! Label the three shapes in Figure $A$ as $(O, M, I)$ and write their exact order in Figure $B$, such as $(I, O, M)$ or $(M, I, O)$. Then substitute the three layers of Figure $C$ into that exact permutation template.
• Coupled Inversion + Side Modification: Often, as the inner polygon expands to the outside, it gains $+1$ side (e.g., an inner triangle becomes an outer square), while the outer polygon shrinks to the inside and loses $-1$ side (e.g., an outer hexagon becomes an inner pentagon).

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3. Shading & Color Negation

Shading transformations in $A : B :: C : ?$ problems follow strict algebraic complementation:

1. Binary Fill Negation: Every black/filled region in $A$ becomes white/hollow in $B$, and every white region in $A$ becomes black in $B$ ($\text{Color}(B) = \overline{\text{Color}(A)}$).
2. Layer-Transfer Shading: The shapes swap positions (inside-outside inversion), and the shading stays fixed to the spatial layer (e.g., "the innermost shape is always black, regardless of which polygon moves inside") versus moving with the polygon itself.
3. Hatching Orientation Flip: Parallel diagonal stripes slanted at $45^\circ$ (rising left-to-right /) flip by $90^\circ$ to $135^\circ$ (falling left-to-right \), or horizontal stripes flip to vertical stripes.

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4. Quantitative Scaling & Proportion Mapping

Some analogies link geometric structure to numerical counts:

• Partition Scaling: Figure $A$ is divided into $k$ equal sectors or internal regions, and Figure $B$ halves ($\frac{k}{2}$) or doubles ($2k$) that partition count.
• Sides-to-Elements Functional Mapping: Figure $A$ is a polygon with $n$ sides and transforms into Figure $B$ containing $f(n)$ internal dots or external line ticks—commonly $n - 1$ dots, $n + 1$ dots, or $2n$ boundary segments. If a 4-sided square maps to 3 internal dots ($n - 1$), then a 6-sided hexagon $C$ maps to $6 - 1 = 5$ internal dots.

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Common Traps: Dihedral Group ($D_4$) Composite Operations

Candidates frequently fall into a classic dihedral symmetry trap when an analogy combines a rotation with a reflection:

The $D_4$ Composition Trap: Suppose $T$ rotates a figure by $90^\circ$ clockwise and then reflects it horizontally across a vertical axis. Many test-takers assume this is equivalent to a $90^\circ$ counterclockwise rotation. It is not! A $90^\circ$ CW rotation followed by a horizontal flip reverses chirality and is mathematically equivalent to a reflection across the anti-diagonal ($y = -x$ where North $\leftrightarrow$ West and South $\leftrightarrow$ East). Always apply the rotation and reflection sequentially to two perpendicular landmarks (such as the top edge and the right edge) to see where both land.

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Worked Step-by-Step Examples

Worked Example 1: Chirality Check in a Rotation + Shading Analogy

Problem Setup:

• Figure $A$: An asymmetric letter P standing upright (vertical spine on the left, curved loop at the top-right), containing a black circle inside the loop and a white triangle below the loop.
• Figure $B$: The letter P is upside down with its vertical spine on the right and its curved loop at the bottom-left, containing a white circle inside the loop and a black triangle above the loop.
• Figure $C$: An upright letter F (vertical spine on the left, two horizontal bars pointing right), with a black square at the top-right and a white star at the bottom-right.
• Question: Find $? = T(C)$.

Step-by-Step Solution:

1. Diagnose Rotation vs. Reflection on $A \to B$: In Figure $A$, the spine of P is on the left (West) and the loop is at the top-right (Northeast). In Figure $B$, the spine is on the right (East) and the loop is at the bottom-left (Southwest). Since both top/bottom (N $\leftrightarrow$ S) and left/right (E $\leftrightarrow$ W) inverted simultaneously, the spatial operator is a $180^\circ$ planar rotation (not a single vertical flip, which would have kept the spine on the left!).
2. Diagnose Shading Rule: The black circle and white triangle in $A$ became a white circle and black triangle in $B$ $\implies$ binary color negation (Black $\leftrightarrow$ White).
3. Apply $T$ to Figure $C$: Rotate the upright F by $180^\circ$: its vertical spine moves to the right, and its two horizontal bars point to the left at the bottom. The top-right black square rotates $180^\circ$ to the bottom-left and negates to a white square. The bottom-right white star rotates $180^\circ$ to the top-left and negates to a black star.

Worked Example 2: 3-Layer Inside-Outside Layer and Fill Permutation

Problem Setup:

• Figure $A$: Outer Triangle (3 sides, white) $\supset$ Middle Square (4 sides, black) $\supset$ Inner Circle (striped).
• Figure $B$: Outer Circle (white) $\supset$ Middle Triangle (3 sides, striped) $\supset$ Inner Square (4 sides, black).
• Figure $C$: Outer Pentagon (5 sides, black) $\supset$ Middle Hexagon (6 sides, striped) $\supset$ Inner Triangle (3 sides, white).
• Question: Find $? = T(C)$.

Step-by-Step Solution:

1. Track Shape Layer Permutation $(O, M, I)$: In $A$, $(O, M, I) = (\text{Triangle}, \text{Square}, \text{Circle})$. In $B$, the new order from outside to inside is $(\text{Circle}, \text{Triangle}, \text{Square}) = (I, O, M)$. That is, the Inner shape moves to Outer, the Outer shape moves to Middle, and the Middle shape moves to Inner.
2. Track Shading Attachment: Compare the fill states $(O_{\text{fill}}, M_{\text{fill}}, I_{\text{fill}})$: in $A$ they are $(\text{White}, \text{Black}, \text{Striped})$; in $B$ they are $(\text{White}, \text{Striped}, \text{Black})$. Notice that the Outer layer stays its original color ($\text{White}$), while the Middle and Inner fills swap! Alternatively, check what happened to each shape's own fill: the Square stayed Black, while the Triangle and Circle swapped their fills ($O_{\text{fill}} \leftrightarrow I_{\text{fill}}$).
3. Apply to $C$: Using $(O, M, I) \mapsto (I, O, M)$ on $C = (\text{Pentagon}, \text{Hexagon}, \text{Triangle})$ gives Outer Triangle $\supset$ Middle Pentagon $\supset$ Inner Hexagon, with $O_{\text{fill}}$ and $I_{\text{fill}}$ swapped so Outer Triangle is black, Middle Pentagon is white, and Inner Hexagon stays striped.

Review the Method

Never confuse composite dihedral D_4 operations: rotating 90° clockwise and then reflecting horizontally produces a diagonal reflection, not a 90° counterclockwise rotation