Dissections of a metal rectangle
In the present popular science paper the following geometric questions are answered: - Which rectangles can be dissected into squares? - When a square can be dissected into rectangles similar to a given rectangle? The proofs are based on a physical interpretation using electrical networks. Only secondary school background is assumed in the paper.
💡 Research Summary
The paper tackles two classic tiling problems—when a rectangle can be dissected entirely into squares, and when a square can be tiled by rectangles that are all similar to a given rectangle—by translating the geometric configurations into electrical networks. The author begins by assigning the rectangle’s horizontal and vertical side lengths (R and C) to the voltage and current of a simple resistive circuit. Each square piece of the dissection corresponds to a unit resistor of equal resistance. By applying Kirchhoff’s laws, the equivalent resistance of the whole network can be expressed in terms of R and C; specifically, the ratio R/C appears directly in the formula. A well‑known result from circuit theory states that an equivalent resistance that is a rational number can be achieved using only series and parallel connections of equal resistors. Consequently, the rectangle can be tiled by squares if and only if the aspect ratio R/C is rational. This yields the first main theorem: A rectangle is dissectable into squares exactly when its side‑ratio is a rational number.
The second problem is approached in a parallel fashion. Suppose a unit square is to be filled with n rectangles, each similar to a given rectangle whose aspect ratio is α (the same for all pieces). The area condition forces the sum of the areas of the α‑scaled rectangles to equal 1. Interpreting each similar rectangle as a resistor whose resistance is proportional to α, the tiling becomes a network of identical resistors arranged in some series‑parallel configuration. The total resistance of such a network is a rational multiple of the single‑resistor value only when α can be written as √q, where q is a rational number. Hence the square can be tiled by rectangles similar to a given rectangle precisely when the rectangle’s aspect ratio is the square root of a rational number. This constitutes the second main theorem: A square can be tiled by rectangles similar to a given rectangle iff the rectangle’s side‑ratio is √(rational).
The central insight is the bijection between the conservation of electric current (Kirchhoff’s current law) and the conservation of area in a dissection, and between voltage drop and the geometric aspect ratio. By replacing intricate geometric arguments with elementary circuit analysis, the proofs become accessible to students with only secondary‑school mathematics (fractions, ratios, basic algebra) and a minimal understanding of series‑parallel resistor networks.
To reinforce the theory, the author supplies a hands‑on classroom experiment: using paper to draw the shapes, copper tape or simple wires to represent resistors, and a battery with a light‑bulb to measure the equivalent resistance. By varying the layout and observing whether the measured resistance is rational, students can directly verify the rational‑ratio condition for square‑tilings and the √‑rational condition for rectangle‑tilings.
Beyond the educational demonstration, the paper situates its results within the broader context of tiling theory and electrical network theory, showing that many results traditionally proved by algebraic topology or advanced combinatorics can be derived from elementary physics. This cross‑disciplinary perspective not only simplifies the proofs but also offers a compelling example of how physical intuition can illuminate pure mathematical problems.
In conclusion, the work delivers two clean, necessary‑and‑sufficient criteria for the two tiling problems, grounded in the simple physics of resistive circuits. It opens a pathway for integrating geometry, algebra, and physics in secondary education, encouraging students to view mathematical structures through the lens of real‑world phenomena and to develop a more unified understanding of STEM concepts.