Infinite Grid of Resistors

Math vs Engineering Perspectives

• Several comments contrast mathematical comfort with infinities versus engineering skepticism.
• Engineers emphasize measurability: you must “apply current” and worry about when steady state is reached, given propagation delays, inductance, and capacitance.
• Others defend the problem as a pure, idealized math exercise using ideal components, distinct from real circuits.

Physical Realism: Quantum, Relativity, Cosmology

• Some suggest quantum discreteness (individual electrons) would limit the “effective size” of the infinite grid; others counter that electron wave behavior and averages make the classical result still meaningful.
• One argument says resistance is inherently dissipative, so there is no coherent quantum path interference.
• There’s an extended GR/cosmology tangent: an infinite “universe” of resistors might be gravitationally unstable (Jeans instability), fragment under expansion-like effects, or collapse into black holes, with discussion of expansion scalars, Raychaudhuri-like equations, and analogies to cosmological models.

Symmetry, Superposition, and Mathematical Structure

• Questions arise about why only two current values (α and β) appear in the symmetry solution. Responses explain starting with 12 independent currents and then enforcing all rotation/reflection symmetries, which group them into two equivalence classes.
• Superposition is justified by linearity of Maxwell’s equations: fields and potentials add, so the two-node solution is the sum of single-node solutions.
• The problem is linked to 2D random walks and to Chebyshev polynomials appearing in the integral solution.

Educational Value and Pedagogy

• Some recall this as a “hated” but favorite exam question; others argue such puzzles overemphasize contrived math over practical skills.
• Defenders say these problems:
  • Train general problem-solving, not rote formula use.
  • Challenge and filter students.
  • Build intuition for symmetry, linear systems, and infinite limits.
• Others note it’s usually a side/bonus topic, not core EE curriculum.

Practical Analogues and Approximations

• Finite resistor networks (including large grids) are genuinely useful; tools exist that solve them exactly via star–mesh transforms.
• Several point out that silicon substrate behavior and sheet resistance in ICs are well-modeled as effectively infinite resistor grids at local scales.
• Suggestions include approximating the infinite grid by measuring large finite grids (e.g., ~100×100) and curve-fitting.

Related Side Discussions

• Some argue the bulk formula (R = \rho l / A) with both (l) and (A) infinite makes the setup “silly”; others implicitly reject this as oversimplified.
• There is a long subthread on why resistors necessarily dissipate energy as heat, and why “current-limiting without heat” is essentially a different kind of device (e.g., switched-mode converters rather than pure resistors).