Overview
Studying at a desk under a single lamp is common practice, especially during long study sessions — but how good is that lighting actually across the whole work surface? This exploration builds a mathematical model of how light spreads from a point source, derives a closed-form illumination function using the inverse square law, then tests that model against 180 real lux measurements taken across a student desk to map where light is strong and where it drops off.
Deriving the Illumination Function
Modeling the light source as spreading uniformly over a sphere of radius \(r\), the surface area at any distance is \(A = 4\pi r^2\). Dividing luminous flux (\(\Phi\)) by that area gives illuminance:
\[E = \frac{\Phi}{4\pi r^2}\]
Substituting the definition of luminous intensity (\(I = \Phi/\omega\)) and assuming a fully “direct” light source (all flux directed downward, \(\omega = 4\pi\) steradians) simplifies this to:
\[E = \frac{I}{r^2}\]This is the working formula used throughout: illuminance at a point equals the source’s luminous intensity divided by the square of the distance to that point.
Mapping the Desk in 3D
To apply the formula to a real desk, each measurement point was treated as a coordinate \((x, y, 0)\) on the desk plane, with the light source fixed above it at \((x_{light}, y_{light}, z_{light})\). Using the Euclidean distance formula, the illumination function becomes:
\[E(x,y) = \frac{I}{(x-x_{light})^2 + (y-y_{light})^2 + z_{light}^2}\]
Results
180 lux readings were taken across the desk at 5cm intervals and plotted as a heatmap:
Setting the partial derivatives \(\partial E/\partial x\) and \(\partial E/\partial y\) to zero shows the function’s critical point occurs exactly at \((x_{light}, y_{light})\) — the point directly beneath the light. This matched the heatmap: the brightest region sat directly under the lamp, falling off sharply toward the desk’s edges, consistent with the inverse-square relationship.
Limitations
The model only accounts for direct light from the source — it ignores reflection off walls, absorption by objects, and light spreading beyond the desk surface. The lux meter used was also limited to two significant figures, and the 180-point grid, while dense, doesn’t capture the theoretically continuous illumination surface.
Takeaway
A desk lit by a single overhead point source loses illumination quickly with distance — and since a desk’s most-used areas often aren’t centered directly under the light, a meaningful fraction of usable light is wasted. The exercise turned an everyday annoyance (uneven desk lighting) into a closed-form, testable multivariable function, and confirmed the inverse square law’s prediction with real measurements.
Python used to generate the heatmap: ```python import numpy as np import matplotlib.pyplot as plt import seaborn as sns
data = np.array([ [15, 17, 19, 22, 25, 28, 31, 34, 36, 36, 37, 37, 37, 35, 33, 30, 28, 25], [15, 18, 20, 23, 26, 30, 33, 35, 37, 37, 38, 38, 38, 37, 35, 32, 29, 26], # …remaining rows omitted for brevity… ])
plt.figure(figsize=(10, 8)) sns.heatmap(data, annot=True, cmap=”coolwarm”, fmt=”.1f”, linewidths=0.5) plt.title(“Heatmap with Data Values (lux)”) plt.show()