II.4. Measurements of Dimension

Because metal and wax both shrink slightly as they cool from molten to solid states, precise measurement of dimension may assist in identifying contemporaneous versions from the same model, or later after-casts or copies (fig. 1). In theory, the former should be nearly identical in size, while the latter should be smaller.

For this type of analysis, it is important to focus on so-called internal measurements, as coined by Ann Allison in her study of bronzes by the Renaissance sculptor Antico (Italian, ca. 1455–1528).² “Internal” refers to measurements taken within areas that do not contain wax-to-wax or metal-to-metal joints, and so should not (in theory) be subject to significant distortion due to assembly of the model or casting. Allison used internal measurements such as width across the chest, length from chin to crown of head, eye to eye, and so on to assist in establishing a “family tree” of different versions cast after the same model (and potentially from the same molds).

Even when internal measurements are used, caution should be exercised. Shrinkage is a complex physical phenomenon that is difficult to model or predict with accuracy, particularly for elaborate hollow forms. Clearly, a variety of processes other than shrinkage can affect final dimensions, including steps taken during model or inter-model production as well as cold work undertaken after casting.³ For instance, it is important to consider that wax models can be bent or distorted (intentionally or unintentionally) prior to investment, resulting in castings that are deformed in comparison to the artist’s original model. In contrast, sand casting (normally using a rigid pattern) tends to produce a more accurate and undistorted casting.

When evaluating published literature, it is important to distinguish between volumetric shrinkage and linear shrinkage. In the technical literature, volumetric shrinkage usually refers to the change in volume of a solid mass upon cooling; this may have a complex and unpredictable relationship with the change in volume of a hollow sculpture or wax model. Linear shrinkage normally refers to the change in exterior dimension of a solid mass of material on cooling; again, the relationship with actual shrinkage of a hollow sculpture or wax model may not be straightforward. For waxes and copper alloys, linear shrinkage should theoretically be about one-third the magnitude of volumetric shrinkage.

Commonly cited estimates of shrinkage suggest that wax, poured as liquid into hard molds, will shrink around 2.5–3% in linear dimension.⁴ During metal casting, copper alloys shrink somewhat less and can be expected to shrink linearly on cooling by around 1–1.5% (fig. 41).⁵ In general, sand- and gypsum-plaster-based investment molds will not shrink appreciably after forming (plaster actually expands slightly on hardening); ceramic shell molds may shrink around 0.5% when fired;⁶ and clay molds (rarely encountered) can shrink dramatically, 4–5% or more in linear dimension.

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Open viewer Figure 41

Measurement can also help determine how versions of the same model were assembled by highlighting differences in the relative orientation of different sections (fig. 76). This can provide insight into original fabrication techniques as well as into the relationships between different versions.⁷ Measurements may also be useful in puzzling out the correct assembly of fragmentary pieces of sculptures found in archaeological contexts.⁸

Open viewer Figure 76

The maximum height of a sculpture resting on a flat surface such as a pedestal or the floor is easily determined by positioning a level (also known as a spirit level or bubble level) on its uppermost tip and establishing the distance between the level and the base by means of a ruler or rigid tape measure. In the case of a figure with an upraised arm or other feature that is higher than the head, measurement should be taken from there. Alternatively, two straightedges may be used together with a tape measure (fig. 407). More than one person may be required to take the measurement depending on the scale of the object.

If a sculpture is mounted on a separately made base (as opposed to an integrally cast one), it is standard practice to provide two height measurements: one with the base and one without. The same measurement process should be used as described above. If pins, sprues, or other elements of the casting extend below the visible sculpture (into the base), the nature of the height measurement should be clearly described in text to avoid confusion.

Open viewer Figure 407

When measuring a sculpture in the round, it is not always clear what constitutes the “front.” In cases where the sculpture has an integrally cast rectangular base, one of its straight edges may help to determine this. The width and depth may be thought of as the internal dimensions of a box (whose sides are parallel to the base) into which the sculpture will fit. The width (sometimes referred to as length) is designated as the side-to-side measurement. The depth is the longest front-to-back measurement (fig. 408). Tools such as a pair of framing squares and a tape measure can be helpful (fig. 409). If the base is round or irregularly shaped, or if there is no base, it will be important to document which side of the sculpture is being considered the front for purposes of the examination. The “internal box measurement” process applies in these cases. Where the integral base extends beyond the sculpture, a decision needs to be made whether this element is to be considered part of the sculpture and thus included in the width and depth measurements. It may be desirable to report dimensions for base and sculpture separately. Keep diagrams of where measurements were taken as part of the sculpture’s documentation.

Measuring a fragmentary bronze presents challenges, especially if it is not clear what the piece’s original position was intended to be. Its incomplete state may also get in the way of adequately comparing its measurements with those of a related cast (fig. 410). Annotated visuals are particularly important in such cases to help describe where the measurements were taken.

Open viewer Figure 408

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Open viewer Figure 410

So-called internal measurements (see II.4§1.1.1 above) can be made manually using calipers, a measuring tape, or a string (fig. 366). Digital calipers now often have direct recording capacities via connection to a computer. Imprints of specific features using plaster or silicone casts may prove useful for comparison of measurements as well, and are quite easy to carry out (fig. 411).

Open viewer Figure 366

Open viewer Figure 411

3D computer modeling offers a more sophisticated means of making measurements (fig. 379), and in particular of comparing measurements across similar casts (fig. 378). A variety of software exists that can assist in identifying innumerable points of comparison in an intricate web.¹⁷ While requiring a certain degree of technical and scientific skill, especially for quantitative calculations, 3D is becoming increasingly user-friendly and affordable.

Three main methods are currently in use for 3D modeling: photogrammetry, structured light scanning, and laser scanning. For basic descriptions of these methods, see II.2§5. There are no hard rules to determine which method is best for creating suitable 3D models for bronze sculpture. The measurement precision of each method can vary tremendously depending on the specific equipment used, the experience of the user, and the nature of the subject. Photogrammetry, for instance, depends heavily on the camera resolution and the software, but even with a 6–7 megapixel camera, Anestis Koutsoudis et al. showed that skilled practitioners can produce models accurate to better than ±1% for measurements 25–120 cm.¹⁸ Professional-grade structured light and laser scanners vary in precision according to manufacturer and model and usually have a specific size range for which they are optimized to scan. Within the target range, precision of both methods can be very high, with accuracies better than the ±0.01% claimed by manufacturers.

In the end, the best method of 3D modeling for any given examination will depend on the desired end use of the measurements (how much precision is necessary) as well as the available equipment and expertise. In practice, most studies on objects in the cultural heritage realm have used either photogrammetry or structured light scanners, while laser scanning has been dominant for sites and architectural applications.

Open viewer Figure 379

Open viewer Figure 378

The cost of the digital camera used in photogrammetry can, of course, vary tremendously depending on resolution, lens quality, and features. Software packages to extract 3D data are evolving and improving rapidly, and several universities are working on 3D reconstruction algorithms.¹⁹ The cost of photogrammetry software depends on one’s needs and output expectations. Free software is very easy to use but offers relatively low quality in regards to point accuracy and overall geometric and visual quality. Professional software can cost up to US$4,000 and allows for the greatest possible quality available but requires high-end computing power and technical expertise to achieve quality results.

Consumer-grade structured light scanners are available for less than US$1,000, but professional-grade scanners more appropriate for technical study usually cost between US$15,000 and $50,000. Operating software (which may require an annual license fee) is normally provided with the scanner, but other software may be desired for additional model manipulation or comparison purposes. This can range from free and open-source packages such as Meshlab or Blender to proprietary packages costing more than US$10,000 and/or requiring annual license fees of many hundreds of dollars.

In the late 2010s, the price of laser scanners certified for metrology ranged from about US$15,000 to $150,000 depending on the resolution and the precision. In addition, proprietary software associated with the scanners can cost from US$10,000 to $50,000, though some free or low-cost data processing software is available.

The speed of data acquisition and processing has increased dramatically in recent years. Using any of the three methods discussed above, an experienced operator can normally accomplish standard-resolution scanning/imaging of a one-meter-high bronze sculpture and generate a finished 3D model in half a day.

In instances where there is direct access to both inner and outer walls of the cast, outside calipers (sometimes called external calipers) may be used to make direct measurement of wall thickness. Electronic outside caliper gauges can be useful, as they give an electronic readout of the dimension in real time and do not require that the caliper be removed to record the measurement. Depending on size, calipers do not allow measurement very far away from the access zone, thus limiting the regions that can be measured. In addition, features such as casting defects, repairs, and corrosion may alter the original thickness in the area where the measurement is carried out. In the end, the representativeness of the measurement may be difficult to assess.

Accurate measurement by radiography can be challenging for two key reasons. First, divergence of the X-ray beam inevitably results in some magnification of the image on the sensor, requiring careful calibration of the image using geometric calculation or reference scales placed in the image. Second, the walls of a sculpture are often curved, which can lead to indistinct edges in X-radiographs.

To complicate matters further, the exterior edge of a roughly cylindrical metal form will present the thinnest cross section in a radiograph. This means that the surface contour of a sculpture may be easily overexposed, thus disappearing in the image and giving the impression of a thinner wall than actually exists (fig. 63). Measurement by X-ray or gamma-ray tomography will generally be more reliable (figs. 379, 398), though precision of the measurement will depend greatly on the equipment used (see II.3).

Open viewer Figure 63

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Open viewer Figure 398

Ultrasonic testing (UT) has proved an efficient method of measuring the thickness of metal. Although used widely in industry, it has only rarely been applied to bronze sculpture (figs. 230, 231).²⁰ UT has the advantage that no dangerous radiation is produced, facilitating on-site and in-lab analysis. UT can also measure very thick metal that might be difficult or impossible to measure with standard X-ray equipment. Unfortunately, UT requires the application of coupling gels to the surface of the object to be inspected, and for any loose paint or corrosion to be removed. In addition, UT signals from metal walls of irregular thickness or with rough surfaces are difficult to interpret, requiring careful attention from experienced technicians.

Open viewer Figure 230

Open viewer Figure 231

To the extent that the interior of a sculpture is accessible for 3D scanning or photogrammetry, it may be possible to merge interior and exterior surface models and create an accurate model of the wall thickness for at least a portion of a hollow sculpture.

Once wall thickness measurements are made, reporting and interpreting results may not be straightforward. Wall thickness will almost certainly vary considerably from area to area. What, then, is the wall thickness to report? Of course, all measurements that have been made should be recorded individually, but for the purposes of characterizing a bronze and comparing it with other castings, providing some summary value(s) is necessary. Unfortunately, but unavoidably, this is bound to be a somewhat subjective process.

First of all, it may be useful to distinguish the different elements of a casting (separately prepared sections of the wax model, for instance, or separately cast sections) and treat them individually. If many measurements are possible for any given element, calculating the median value may be a useful way of characterizing the thickness. If not, then it may be up to an investigator’s judgment to choose a “typical” thickness measurement to report. Some researchers suggest that investigators attempt to determine a “targeted thickness” representing the founder’s envisioned thickness,²¹ though this has not yet been widely accepted.

In addition to reporting quantitative results, some qualitative descriptors of a sculpture’s thickness may be useful. “Homogeneous,” “regular,” and “conformal” are all terms that convey important information about wall thickness, though their definitions are imprecise. The thickness of the wall may be said to be homogeneous if it is more or less the same throughout the sculpture. The thickness may be termed regular if it does not show any sharp discontinuities or changes (as observed at wax-to-wax joints). The metal wall is said to be conformal if the inner and outer contours match closely such that the thickness remains constant as the form meanders.