The best sensor is looking sideways

A small exploration robot has wandered beneath a sofa in a cluttered living room.

Its first sensor is an ultra-precise laser rangefinder pointing due east. It measures the robot’s east-west position with sub-millimeter precision.

Its second sensor is another identical laser rangefinder, also pointing due east. It confirms the first reading with pristine accuracy.

The robot’s engineers celebrate: both sensors agree down to the last decimal place!

Yet when you ask where the robot is located in the room, a sobering fact emerges: the robot could be anywhere along a ten-meter corridor stretching from north to south. It knows its east-west coordinate with perfection, but it remains completely blind to where it sits along the room’s length.

Now add a third sensor: a crude, noisy ultrasound transducer pointing due north.

Its precision is ten times worse than the lasers. Yet the moment its reading arrives, the ten-meter corridor instantly collapses into a tight, two-centimeter uncertainty circle.

The cheap sensor won not by being sharp, but by looking sideways.

When I brought these engineering dilemmas into Epistemic Physics, they illuminated an essential truth: information is fundamentally geometric. You can accumulate an enormous budget of precision, but if all your measurements point in the same direction, your epistemic mass remains zero.

Let us explore how the spacetime cone governs sensor fusion, mechanical spring networks, and the mathematics of curiosity.

The measurement stripe: Fisher information as curvature

Let us formalize the robot’s localization problem in the two-dimensional plane.

Represent the robot’s unknown true position by a vector $\mathbf{z}=(z_1,z_2{)}^{\mathsf{T}}$.

Each sensor $i$ measures position projected along a unit directional axis ${\mathbf{a}}_i=(\cos {\theta }_i,\sin {\theta }_i{)}^{\mathsf{T}}$, where ${\theta }_i$ is the angle of the sensor’s line of sight.

The scalar measurement recorded by sensor $i$ is corrupted by additive Gaussian noise ${\epsilon }_i$:

$$y_i={\mathbf{a}}_i^{\mathsf{T}}\mathbf{z}+{\epsilon }_i=z_1\cos {\theta }_i+z_2\sin {\theta }_i+{\epsilon }_i,$$

where ${\epsilon }_i\sim \mathcal{N}(0,{\sigma }_i^2)$ has mean zero and variance ${\sigma }_i^2$.

A single sensor does not identify a point: it constrains the robot to a stripe perpendicular to ${\mathbf{a}}_i$.

Define the precision of sensor $i$ as the inverse variance of its noise:

$$w_i=\frac{1}{{\sigma }_i^2}.$$

To combine multiple sensor readings, we minimize the weighted sum of squared residuals:

$$Q(\mathbf{z})=\frac{1}{2}\sum _{i=1}^M{w}_i(y_i-{\mathbf{a}}_i^{\mathsf{T}}\mathbf{z}{)}^2.$$

Near its minimum, this quadratic objective function forms a parabolic bowl. The curvature of this bowl is given by the Hessian matrix, known in statistics as the Fisher information matrix $F$:

$$F={\mathrm{\nabla }}^{2}Q(\mathbf{z})=\sum _{i=1}^M{w}_i{\mathbf{a}}_i{\mathbf{a}}_i^{\mathsf{T}}.$$

The outer product ${\mathbf{a}}_i{\mathbf{a}}_i^{\mathsf{T}}$ is a rank-one projection matrix:

If $F$ is invertible, the covariance matrix of the best linear unbiased estimator is:

$$\mathrm{\Sigma }=F^{-1}.$$

Directions where the information bowl curves steeply upward correspond to small estimation errors; flat directions correspond to unobserved blind spots.

Doubling the angle: Why lines have no arrows

Before summing our information matrices, our robot poses a subtle geometric puzzle.

Suppose sensor 1 points due east (${\theta }_1=0^{\circ }$), measuring $z_1$.

Suppose sensor 2 points due west (${\theta }_2={180}^{\circ }$). Its directional vector is ${\mathbf{a}}_2=(-1,0{)}^{\mathsf{T}}$, and its measurement is $y_2=-z_1+{\epsilon }_2$.

As directional vectors, ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ point in opposite directions (${\mathbf{a}}_1=-{\mathbf{a}}_2$).

Yet in the information matrix:

$${\mathbf{a}}_2{\mathbf{a}}_2^{\mathsf{T}}=(-{\mathbf{a}}_1)(-{\mathbf{a}}_1{)}^{\mathsf{T}}={\mathbf{a}}_1{\mathbf{a}}_1^{\mathsf{T}}.$$

A sensor looking west imposes the exact same geometric constraint as a sensor looking east! A measurement axis is an unoriented projective line, not a directed vector.

To give these lines an invariant coordinate system where identical lines correspond to identical points, we double the angle:

$${\tilde{\mathbf{n}}}_i=(\cos 2{\theta }_i,\sin 2{\theta }_i).$$

Look at what this double-angle map accomplishes:

• East ($\theta =0^{\circ }$) and west ($\theta ={180}^{\circ }$) both map to $2\theta =0^{\circ }$: they become identical points on the circle!
• North ($\theta ={90}^{\circ }$) maps to $2\theta ={180}^{\circ }$: perpendicular measurement axes become diametrically opposite points!

The doubled angle turns perpendicularity into mutual cancellation.

The spacetime cone of directional sensing: Line mass as determinant

Now let us sum the precision budget across all sensors.

Using the trigonometric double-angle identities ${\cos }^2\theta =\frac{1+\cos 2\theta }{2}$, ${\sin }^2\theta =\frac{1-\cos 2\theta }{2}$, and $\cos \theta \sin \theta =\frac{\sin 2\theta }{2}$:

Summing over all sensors $i=1,\dots ,M$:

where we define the total precision $\tau$ and the doubled-angle directional moments $C$ and $S$:

$$\tau =\sum _{i=1}^M{w}_i,C=\sum _{i=1}^M{w}_i\cos 2{\theta }_i,S=\sum _{i=1}^M{w}_i\sin 2{\theta }_i.$$

Now compute the determinant of the information matrix $F$:

$$detF=\frac{1}{4}[(\tau +C)(\tau -C)-S^2]=\frac{1}{4}[{\tau }^2-C^2-S^2]=\frac{{\tau }^2-r^2}{4},$$

where $r=\sqrt{C^2+S^2}\ge 0$ is the magnitude of the directional anisotropy.

Multiplying by 4 reveals the invariant line mass of the sensor array:

$$\boxed{4detF={\tau }^2-(C^2+S^2)={\tau }^2-r^2=m_{\mathrm{line}}^2.}$$

Look at that equation:

This is the $(2+1)$-dimensional Minkowski spacetime interval!

• Total precision $\tau$ acts as forward-ticking epistemic time.
• Directional moments $(C,S)$ act as spatial momentum vectors.
• Four times the determinant of the information matrix is the invariant relativistic rest mass squared!

Consider what this physical correspondence tells us:

1. Parallel redundancy ($r=\tau$): If all sensors look along the same axis (say, all at $\theta =0^{\circ }$), then $C=\tau$ and $S=0$. The spatial momentum matches time: $r=\tau$. The invariant mass vanishes: $m_{\mathrm{line}}=0$. The array is lightlike: $detF=0$. The information matrix is singular, leaving an infinite unobserved blind corridor.
2. Orthogonal balance ($r=0$): If the precision budget is divided equally between perpendicular axes (say, $\tau /2$ at $0^{\circ }$ and $\tau /2$ at ${90}^{\circ }$), then $C=(\tau /2)(1)+(\tau /2)(-1)=0$ and $S=0$. The spatial momentum vanishes: $r=0$. The line mass reaches its theoretical maximum: $m_{\mathrm{line}}=\tau$. The determinant is maximized: $detF={\tau }^2/4$.

Orthogonal sensors cancel their directional anisotropy, maximizing the invariant epistemic mass of the sensor array.

Find the robot under the sofa

Give the second sensor a new question.

Sensor A measures east-west. Rotate sensor B and divide a fixed precision budget between them. Drag the teal handle to rotate B, or use the angle control. Their measurement noises are independent.

Angle of sensor B

60°

Precision assigned to A

50%

Sensor precision budget

8 units

Parallel Perpendicular Opposite

Add one total unit of isotropic prior precision

Strongest precision

6.00

Weakest precision

2.00

Condition number

3.00

Cone quantity $m_{\mathrm{line}}$

6.93

$$4detF={\tau }^2-C^2-S^2$$

In the interactive laboratory above, rotate the sensors and observe the uncertainty ellipse. Watch how the ellipse area contracts to a minimum when the sensors are perpendicular (${90}^{\circ }$), and stretches into an unbounded strip when they align ($0^{\circ }$ or ${180}^{\circ }$).

The skinny ellipse: Condition numbers and rapidity

The eigenvalues of the information matrix $F$ describe the strongest and weakest directions of precision in the plane:

$${\lambda }_{\pm }=\frac{\tau \pm r}{2}.$$

The unit uncertainty contour is an ellipse whose semiaxes are $1/\sqrt{{\lambda }_{+}}$ and $1/\sqrt{{\lambda }_{-}}$. The area of this uncertainty ellipse is inversely proportional to the square root of the determinant:

$$\text{Area}=\frac{\pi }{\sqrt{detF}}=\frac{2\pi }{\sqrt{{\tau }^2-r^2}}=\frac{2\pi }{m_{\mathrm{line}}}.$$

Maximizing line mass minimizes the area of the uncertainty patch!

Now define the fractional anisotropy ratio:

$$q=\frac{r}{\tau }\in [0,1).$$

The condition number of the information matrix, which measures how skinny and ill-conditioned the ellipse is, evaluates to:

$$\kappa =\frac{{\lambda }_{+}}{{\lambda }_{-}}=\frac{\tau +r}{\tau -r}=\frac{1+q}{1-q}.$$

Now recall our epistemic rapidity coordinate from relativity:

$$\rho =\mathrm{artanh}(q)=\frac{1}{2}\ln \left(\frac{1+q}{1-q}\right).$$

Taking the exponential of both sides:

$$\boxed{\kappa =e^{2\rho }.}$$

The condition number of a sensor array is the exponential of twice its epistemic rapidity!

When an array is well-balanced ($\rho =0$), the condition number is $\kappa =1$: the uncertainty patch is a perfect circle. As the measurements become lopsided ($\rho \to \mathrm{\infty }$), the condition number explodes exponentially, stretching the ellipse into an infinitely needle-thin stripe.

Mechanical duality: Sensors as elastic springs

We can build an exact mechanical duplicate of this sensor network using simple physics laboratory equipment.

Replace every sensor with an ideal mechanical spring.

Let the spring be anchored along the sensor’s line of sight ${\mathbf{a}}_i$, with spring constant (stiffness) $k_i=w_i$.

When the central body is displaced by a small vector $\mathrm{\Delta }\mathbf{z}$, the extension of spring $i$ is $\mathrm{\Delta }{\ell }_i={\mathbf{a}}_i^{\mathsf{T}}\mathrm{\Delta }\mathbf{z}$.

The elastic potential energy stored in the entire spring system is:

$$U(\mathrm{\Delta }\mathbf{z})=\frac{1}{2}\sum _{i=1}^M{k}_i(\mathrm{\Delta }{\ell }_i{)}^2=\frac{1}{2}\sum _{i=1}^M{w}_i({\mathbf{a}}_i^{\mathsf{T}}\mathrm{\Delta }\mathbf{z}{)}^2=\frac{1}{2}\mathrm{\Delta }{\mathbf{z}}^{\mathsf{T}}\left(\sum _{i=1}^M{w}_i{\mathbf{a}}_i{\mathbf{a}}_i^{\mathsf{T}}\right)\mathrm{\Delta }\mathbf{z}=\frac{1}{2}\mathrm{\Delta }{\mathbf{z}}^{\mathsf{T}}F\mathrm{\Delta }\mathbf{z}.$$

The mechanical stiffness matrix of a spring network is mathematically identical to the Fisher information matrix of a sensor network!

Notice the physical intuition this duality gives us:

• If all springs are parallel, the central body is stiff along the spring axis, but completely floppy in the transverse direction: you can slide it sideways with zero resistance. A missing measurement direction and a floppy mechanical direction are twins at the level of these quadratic bowls!
• If the springs are balanced in perpendicular pairs, the central body resists displacement equally in all directions: the mechanical stiffness is isotropic.

Imagine designing a sensing structure by giving it a mechanical model. Soft directions reveal poorly observed motions. An engineer can tug the model, find the wobble, and place a sensor where a supporting spring would go. Larger structures add joints, boundary conditions, and higher dimensions, but the local principle remains delightfully tangible.

What if the robot already has a rough position estimate? A Gaussian prior adds an isotropic baseline: an initial circular puddle of stiffness that supports every direction equally. Turn that prior on while the sensors are parallel, and the infinite stripe instantly becomes a finite, elongated ellipse. Carrying the inherited prior and the fresh sensor observations together allows an autonomous system to distinguish between what was inherited and what was measured.

Color the likelihood, feel the spring

Instead of drawing only one uncertainty contour, color every possible displacement. For zero-valued sensor readings, the relative Gaussian likelihood is $L(\mathbf{d})/L(\mathbf{0})=e^{-Q(\mathbf{d})}$. Choose spring stiffnesses equal to sensor precisions in matching units, and the mechanical energy is $U(\mathbf{d})=Q(\mathbf{d})$. Every contour of one becomes a contour of the other.

Predict what will happen to the map when the sensors become parallel. Then drag the gold probe along the remaining stripe. The color scale measures likelihood relative to the best fit; it stays meaningful even when the stripe cannot be normalized into a position distribution.

A likelihood map and a spring bowl

Where can the robot still be?

Reset map

Two sensors both report zero. Turn sensor B, then drag the gold probe through the possible positions. A matched pair of springs assigns exactly the same quadratic cost to that displacement.

Sensor A precision 1 · broad stripe4 · medium stripe9 · narrow stripe Sensor B precision 1 · broad stripe4 · medium stripe9 · narrow stripe

Point together Look sideways Probe one unit north

Relative Gaussian likelihood L / L_max

0 · low0.51 · best fit

Sensor A axisSensor B axis · 60°Selected displacement

Sensor mismatch

$$Q(\mathbf{d})=\frac{1}{2}{\mathbf{d}}^{\mathsf{T}}F\mathbf{d}$$

Q = 2.691

Higher mismatch makes this position less likely.

Elastic energy

$$U(\mathbf{d})=\frac{1}{2}{\mathbf{d}}^{\mathsf{T}}K\mathbf{d},K=F$$

U = 2.691

Matching spring stiffnesses assign the same cost.

At displacement (0.60, 0.80), both costs are 2.691. The map color is e^−Q = e^−U = 0.068. The low-energy region and the high-likelihood region have the same outline.

Precision is inverse noise variance. Both readings are zero, their Gaussian errors are independent, and the plot covers −2 to 2 position units on both axes. Colors show likelihood relative to its maximum, not probability per cell. In the spring model choose stiffnesses equal to the sensor precisions in matching units. The vector d is the gold displacement from the origin.

Drag B to turn its axis. Focus either handle for arrow keys; Shift moves the gold probe farther.

Pairwise disagreement and the graph Laplacian

Now let us look at oriented evidence vectors ${\mathbf{n}}_i$ of unit length, carrying weights $w_i$.

Compute the total weight $\tau =\sum w_i$ and the resultant vector $\mathbf{x}=\sum w_i{\mathbf{n}}_i$.

Expand the squared Minkowski mass:

$${\tau }^2-\Vert \mathbf{x}{\Vert }^2={\left(\sum _i{w}_i\right)}^2-{\Vert \sum _i{w}_i{\mathbf{n}}_i\Vert }^2=\sum _{i,j}{w}_i{w}_j-\sum _{i,j}{w}_i{w}_j({\mathbf{n}}_i\cdot {\mathbf{n}}_j).$$

Using the identity $1-{\mathbf{n}}_i\cdot {\mathbf{n}}_j=\frac{1}{2}\Vert {\mathbf{n}}_i-{\mathbf{n}}_j{\Vert }^2$:

Look at that identity:

Epistemic rest mass squared is the total pairwise disagreement energy of the network!

In spectral graph theory, if we treat each evidence vector as a node on a complete graph with edge weights $c_{ij}=w_i{w}_j$, this quadratic sum is precisely the Dirichlet energy of the graph Laplacian:

$$\mathcal{E}=\frac{1}{2}\sum _{i,j}{c}_{ij}\Vert {\mathbf{n}}_i-{\mathbf{n}}_j{\Vert }^2.$$

When all nodes agree (${\mathbf{n}}_i={\mathbf{n}}_j$), the Dirichlet energy vanishes: the system has zero mass and zero disagreement.

When nodes hold opposing convictions, the Dirichlet energy climbs, endowing the network with internal tension and epistemic rest mass.

This is the foundational energy behind graph Laplacians, matrices that encode how neighboring values differ across a network. Spielman’s spectral graph theory text develops their deep connections with mechanical springs, electrical resistor networks, and graph geometry.

For a general network, we can choose different connections and weights. Its energy then measures disagreement along those particular relationships: a weather station compares readings with nearby stations; a community model connects people who exchange information. Choosing the graph specifies whose differences interact.

A map that knows where to stay curious

Put these ideas together and a principled learning machine comes into view.

Every cell in its spatial map stores an amount $\tau$, a directional resultant $\mathbf{x}$, and an information matrix describing its strongest and weakest constraints. Neighboring cells exchange information along a graph. The machine directs new measurements toward weakly supported directions and regions where neighboring accounts disagree.

Consider a robot reaching the edge of an open doorway: it should preserve the sharp physical difference between solid wall and empty passage. A naive network that simply minimizes every disagreement would smear that boundary away through Laplacian diffusion. Its graph must instead learn when an edge should transmit a smoothing correction and when it should preserve a vital physical distinction.

We can make the idea concrete with our sofa robot. Give it a fixed measurement budget. Compare three autonomous policies:

1. The confirmation trap (greedy precision): The robot repeatedly fires its sharpest east-west sensor. Its total precision $\tau$ climbs rapidly, but its line mass $m_{\mathrm{line}}$ and weak eigenvalue ${\lambda }_{-}$ remain frozen at zero. It revisits what it already understands and learns nothing about north-south obstacles.
2. The isotropic policy: The robot rotates its sensors uniformly, spreading its budget evenly. This guarantees convergence, but wastes half its energy refining directions that are already well-constrained.
3. The active epistemic policy (curiosity targeting the floppy mode): The robot computes the eigenvectors of its current information matrix $F$. It identifies the weakest eigenvalue ${\lambda }_{-}$: the floppy direction where the uncertainty ellipse is widest. It then aims its next sensor directly along that weak axis.

Track the robot’s position error, uncertainty ellipse area, and information gain across steps. By aiming directly at its own blind spot, the robot collapses its uncertainty ellipse in the minimum possible number of trials.

Give three robots the same allowance

Here is a controlled version of that race. All three robots begin with the same Gaussian estimate, and every fresh reading costs the same and contributes one unit of precision. The available measurement axes are east-west and north-south. Their total precision grows equally; their predicted uncertainty ellipses take different paths.

Compare repeating the east-west measurement, alternating axes, and always choosing the weaker axis. Scrub the step dot to put all three at the same budget. The weak-axis rule is optimal for the determinant among allocations of whole readings to these two available axes; the displayed curves let us see exactly what that guarantee buys.

Three policies, one measurement budget

Spend the next reading on a blind spot.

Start over

Three copies of the robot inherit the same position estimate. One keeps measuring east-west, one alternates axes, and one measures whichever direction is currently weakest. Advance them together and compare their remaining uncertainty.

East already sharp Start balanced Almost blind north

Starting precision: east-west 9, north-south 1. Every new independent reading has unit precision and the same cost.

−4 readings Available budget

16

+4 readings

Repeat eastAlternate axesAim at weakest

One reading back Step

6

/ 16 Take the next reading Spend the full budget

All three position plots have the same scale: ±1.00 units in each direction. Dashed outlines show the starting estimate.

Policy	Readings east / north	Line mass
Repeat east	6 / 0	7.75
Alternate axes	3 / 3	13.86
Aim at weakest	0 / 6	15.87

All three now have total precision 16.00. Aiming at the weakest direction leaves 37.8% of the starting ellipse area. The amount is the same; its directional coverage changes the result.

$$F^{+}=F+\mathbf{a}{\mathbf{a}}^{\mathsf{T}},m_{\mathrm{line}}=2\sqrt{detF},A=\frac{2\pi }{m_{\mathrm{line}}}$$

Here a is the chosen unit measurement direction, and A is ellipse area. The contour is d^TFd = 1, enclosing 39.3% probability for the Gaussian estimate. Among allocations of whole, equal-cost readings to these two axes, choosing the weaker axis maximizes the determinant at each budget. The curves show analytically predicted uncertainty. Drag the teal dot to change steps, or focus it and use the arrow keys.

The cone has become an engineering design tool. Its height measures the measurement budget, its lean measures directional imbalance, and its interior records the coverage created by complementary, orthogonal contributions. Curiosity is the rigorous geometric policy of pointing your next sensor where your current information matrix is flat. Sometimes the fastest route to a better answer is to turn sideways and ask a new question.

In our final excursion, we will examine what happens when observations pass through a partial aperture in When evidence puts on sunglasses, uncovering how projection and marginalization transform belief.