Overview
Hanging picture frames on a wall presents a deceptively simple geometric problem with an elegant mathematical solution. When arranging multiple frames of equal width such that all gaps—between frame edges, between frames and wall edges—are uniform, the spacing between frame centers differs from the distance between wall edges and outer frame centers.
The Problem
Given (Input Parameters):
- wwall = width of the wall
- wframe = width of each frame (all frames have equal width)
- nframe = number of frames to space on the wall
Find (Output Variables):
- dedge = distance between edges of adjacent frames
- dwall-to-center = distance from wall edge to centers of outer frames
- dcenter-to-center = distance between centers of adjacent frames
- xcenteri = position of the center of frame i
The Solution
This work derives a set of analytical equations that solve the centering problem directly. By treating frame count as a control variable and wall/frame widths as boundary conditions, we can compute all required distances from first principles. The solution includes verification equations to confirm measurements, as well as generating functions to locate frame centers and edges.
Key Result
The fundamental insight is that equal spacing between frame edges does not yield equal spacing between frame centers. The distance between consecutive frame centers is larger than the distance from the wall edge to the outermost frame's center, precisely because the outer frames are offset by half their width relative to the wall boundary.
Practical Application
With the derived equations, installing evenly-spaced frames becomes straightforward: measure wall and frame dimensions, apply the formulas, and use the resulting center positions as installation points. This eliminates guesswork and ensures professional results.