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Capillary origami uses surface tension to fold an initially flat elastic plate around a liquid droplet. The fold trades a reduction in surface energy for an increase in bending energy; the initial outline helps select the resulting enclosure.^[ref]

In the 2007 experiments of Charlotte Py and collaborators, 40–80 micrometre polydimethylsiloxane sheets rested on a water-repellent support. A water droplet reached the sheet's corners; time-lapse images followed its evaporation. Squares changed from four-corner bending to a tubular shape; triangles formed a tetrahedron. Smaller or thicker sheets could bend and then reopen without enclosing the droplet.^[ref]

The elastocapillary length makes the thickness dependence quantitative. For fixed Young's modulus, Poisson's ratio and surface tension, the bending stiffness obeys $B\propto h^3$ and ${\ell }_{\mathrm{ec}}\propto h^{3/2}$. Halving thickness therefore reduces ${\ell }_{\mathrm{ec}}$ by $2^{3/2}$, about $2.83$. At the same lateral size $L$, the sheet elastocapillary energy ratio $\eta =\gamma L^2/B$ increases eightfold. This calculation explains why making the sheet thinner can enable wrapping even without changing its outline.

The sheet elastocapillary energy ratio compares available energy scales; it does not determine a unique final shape. The contact angle, initial liquid volume and outline also matter. A cut chosen to permit several panels to meet is a mechanical design choice, while the surface tension supplies the driving force. Predicting the complete folding path requires more than the scalar energy comparison. The same group reported cross-shaped templates forming a cube and flower-shaped templates forming nearly spherical enclosures.^[ref]

Elastocapillarity is the coupling between surface tension and elastic deformation. Changing an interface can reduce surface energy while bending a compliant solid increases bending energy. Thin structures can deform strongly when the available capillary energy competes with their elastic resistance.

For an isotropic thin elastic plate, its bending stiffness iswhere $E$ is Young's modulus, $h$ the thickness and $\nu$ Poisson's ratio. A sheet bent over a size $L$ has curvature of order $1/L$. Its bending energy scales as $B(1/L^2)L^2\sim B$, while the capillary energy scale is $\gamma L^2$, where $\gamma$ is surface tension. Balancing these scales gives the elastocapillary lengthHere $B$ has units of energy and $\gamma$ energy per area, so the ratio has units of length squared. At fixed material properties, ${\ell }_{\mathrm{ec}}\propto h^{3/2}$; doubling thickness multiplies this length by $2\sqrt{2}$. Geometry and contact angle enter the more detailed wrapping criterion.^[ref]

The 2007 capillary origami experiments of Charlotte Py and collaborators provide a direct demonstration. They cut polydimethylsiloxane membranes, 40–80 micrometres thick, into planar shapes, placed them on a water-repellent support, added a water droplet and recorded the deformation during evaporation. Suitable triangular sheets folded around the shrinking drop into a tetrahedral enclosure. Smaller or stiffer sheets failed to close completely. The observations connected a controlled material thickness and initial shape to the competition between surface energy and bending energy.^[ref]

Measurement reproducibility is measurement precision under specified reproducibility conditions of measurement. These include changed locations, operators and measuring systems; the measurement procedures may also differ. The comparison must state what changed and what remained common.^[ref] This is a metrological use of reproducibility, concerned with agreement of measured values under declared conditions.

An original comparison sends a stable reference block to three laboratories. Each laboratory uses its own differential length comparator and operator, measures the same specified measurand, and reports three length errors relative to the common reference, in micrometres:Each laboratory has sample standard deviation $1\mu \mathrm{m}$, yet their arithmetic means differ by as much as $6\mu \mathrm{m}$. Relative to the grand mean $2/3\mu \mathrm{m}$, the total squared deviation separates into within-laboratory and between-laboratory terms:Consequently the nine chosen values have pooled sample standard deviation $\sqrt{62/8}\approx 2.78\mu \mathrm{m}$. This is a descriptive statistic for these observations, not a universal estimator prescribed by the definition of measurement reproducibility. Reporting only the three within-laboratory spreads would conceal the disagreement.

The physical comparison must also monitor transport damage, reference stability and temperature corrections. A changing reference block could imitate a laboratory difference. Measurement reproducibility alone does not establish measurement trueness: all laboratories could share the same erroneous reference value or correction. The changed apparatus and conditions belong alongside the reported statistic.

A non-breaking space is a separating character intended to keep adjacent text together during ordinary line breaking. The common Unicode character is U+00A0 NO-BREAK SPACE. It has line-break class GL; the surrounding rules determine which boundaries are prohibited.^[ref]

For an example, 25\u00a0µm is a visible spelling of an escape notation for a number, U+00A0, and a micrometre symbol. The actual text 25 µm keeps the number and unit together at that separator. Ordinary 25 µm instead uses U+0020 SPACE and can provide a wrap opportunity there. Neither choice fixes the physical width of the gap in every computer font.

The rule is local. In the sequence A\u0020\u00a0B, a break after the ordinary space can still be allowed, leaving the non-breaking space on the following line. Thus inserting U+00A0 does not make an entire text paragraph unbreakable.^[ref]

Positioning repeatability describes the scatter of positions reached when a motion system repeatedly returns to the same commanded position under specified conditions. It is an application of measurement repeatability, a form of measurement precision. A system may return to almost the same wrong position each time, so small scatter does not establish measurement accuracy.^[ref]

The approach direction belongs in the test description. Unidirectional positioning repeatability keeps it fixed; bidirectional positioning repeatability includes returns from both directions. To measure the assembled stage, independently observe its final position rather than only the motor's rotary encoder count. Physik Instrumente describes a static test that lets the platform stop before reading it, using an interferometer for displacement and an autocollimator for angular error, with two reflecting mirrors on the stage.^[ref]

For an original numerical example, repeatedly command $x_{\ast }=10.000\mathrm{mm}$. Suppose an independent position readout with known measurement calibration gives the errors $e_j=x_j-x_{\ast }$, in micrometres,Each direction has the same sample standard deviation,Yet their arithmetic means are 0 and $2.0\mu \mathrm{m}$. The magnitude of their mean difference, a positioning reversal error at this target, is 2.0 micrometres, far larger than the within-direction scatter. Pooling all eight values gives a sample standard deviation of $\sqrt{8.04/7}\approx 1.149\mu \mathrm{m}$; that statistic mixes the directional shift with the smaller return-to-return variation.

The calculation uses a declared sample standard deviation, not an unspecified manufacturer's tolerance or a universal repeatability convention. A positioning specification must state its approach directions, target positions, settling/readout procedure, load, environmental conditions, statistic and measurement uncertainty. A single target does not establish performance over the travel range. The manufacturer's method explicitly separates directional means from scatter and combines information from multiple targets when forming axis-level specifications.^[ref]

Backlash is one possible cause of direction-dependent behavior, while thermal expansion and instrumental drift can move the apparent target over time. A repeatability experiment must distinguish those effects from random scatter. Measuring small returns to one target also answers a different question from minimum incremental motion, which asks whether a small commanded change can actually be executed.

An SI prefix supplies an exact decimal factor when attached to a unit of measurement. For example, milli means $10^{-3}$ and micro means $10^{-6}$; a millimetre and a micrometre are correspondingly scaled metres. The prefix symbol and unit symbol form one unit that can itself be raised to a power.^[ref]

For an original conversion example,The area conversion squares the factor as well as the unit. Applying just one factor of $10^{-3}$ to that area would give $0.004{\mathrm{m}}^2$, a value 1000 times too large. Scientific notation makes the power explicit.

Prefix symbols are case-sensitive and cannot be stacked. For mass, prefixes attach to gram, because the base-unit name kilogram already contains a prefix. SI prefixes denote powers of ten, even in a computing context; they do not silently change into powers of two.^[ref] An exact unit conversion scales any stated measurement uncertainty with the value; it does not make an uncertain measurement exact.