Hypothesis

THE ROMAN DODECAHEDRON AS A THREE DIMENSIONAL RECORDING AND MEASURING INSTRUMENT FOR DOCUMENTING AND DESIGNING THE OPTIMAL SLING AMMO SHAPE FOR MAXIMUM BATTLE EFFECTIVENESS

John W. Ladd

Over the years, 28 theories have been documented, many of them dismissed, and none of them considered satisfactory.  Although some of the proposed uses are certainly possible, the theories fail to explain why a relatively sophisticated device would be used instead of alternative means to successfully achieve the desired function.  On March 9th, 2011, a new theory was developed, and was publicly presented at the University of Michigan, Ann Arbor, on April 25th.  However, as recently as June 10th, 2011, media has reported that "Ancient Dodecahedron's Purpose Remains Secret" (Alexandria Hein, Fox news).

A new hypothesis

This article presents a purposeful use of the dodecahedron, along with historical support.  Video explanation of critical principles is embedded into the article to aid the illustration.  In short description, my theory can be summarized as follows: the dodecahedron was a fluid immersion cage, optimized for the recording of three dimensional shape of military projectiles, particularly suited to sling ammo, in order to improve upon the design and manufacturing of such artillery projectiles to maximize kill ratio.  An extension of Archimedes volumetric displacement principle to additional angles and fine increments mathematically allowed the determination of deviation from production quality and design standards.  The timeline of mathematical development, experimental volume displacement principles, and platonic solid development points to the inventor of the Roman Dodecahedron to have lived around the time of Archimedes, or shortly after.  Facts of the events surrounding Archimedes death support the possibility he was directly involved in the development of the Roman Dodecahedron as a tool to support the design of sling ammo, the longest range means of attack, for the Greek army.

The functioning of the dodecahedron as a measuring device can be understood from the concept of incremental fluid displacement recording.  A simple example, which does not make use of a dodecahedron, but is a simplified case of the underlying concept, is illustrated below (see figure 1), where a sphere is depicted being submerged incrementally with an Erlenmeyer type flask:

hypothesis_1

With incremental recording of the of the displacement of the sphere, in accordance with Archimedes principle, a specific displacement profile is expected to be measured, indicative of how the incremental volume, or "slice", changes as a function of fluid height. The experiment was performed, using the apparatus below, (see figure 2a), consisting of a ruler, a fluid tank, and a glass sphere.  The experimentally tabulated data is plotted against the theoretical expectation (see figure 2b):
hypothesis_2

It was evident that the data recording was in close agreement to what would be expected from Archimedes principle; the shape of the sphere gives the specific profile that is expected.

However, the inverse statement is not general--knowing the specific profile measured in this example does not guarantee that the shape of the object is a sphere; the sphere is only a subset of an infinite number of possible shapes which could give rise to the measured profile in figure 2b.  Understanding this key point is important in my hypothesis of the purpose of the Roman Dodecahedron artifact; the idea for the utility of the device rests on the principle that the specific Roman dodecahedra that have been found were an optimized means to provide a unique dataset for describing the deviations in the shape of a sample from a known standard.  In order for the hypothesis to be reasonable, mathematical and practical measurement solutions to this question of uniqueness must result from the design and measurement method.  In addition, alternative (and industrially cheaper) solutions to this problem must not have been available with technology of the era, or the Romans would not have been justified in using this particular approach for 3D measurement and quality assurance.

The Problem of Uniqueness

The ability of different objects to provide the same displacement profile, and therefore not be unique, is illustrated below (see figure 3), where each object has a similar volume as a function of fluid height, despite having a different shape:
hypothesis_3
In a real physical case, variation can take place among all 3 dimensions, and an infinite number of shapes can potentially create similar measurement profiles.  To improve upon the ability to compare two objects (a test object vs. a standard) for likeness, additional angles of measurement can be introduced (see figure 4):

hypothesis_4
At the additional angle, rotated 45 degrees compared to figure 3, the objects in figure 4 will now provide different displacement profiles.  It is therefore evident that to improve the uniqueness of the dataset, additional angles and/or increased number of measurement increments per angle can be used.  However, the object must be supported by a cage to control the angle of measurement and to physically support the test subject to prevent it from rolling over.

The Need for Angular Support of Test Subject

One possible choice of immersion cage would be a dodecahedron structure.  In this example, the test subject is an ellipsoid (see figure 5):

hypothesis_5


The supporting cage allows the test subject to be immersed at a variety of angles.  How the hole patterns allow for resolution read-points (6 unique angles corresponding to 12 faces for a dodecahedron) of the displacement profile will be discussed.  In order to understand the ideal measurement strategy for utilization of the hole patterns, the mathematics of uniqueness should be considered (next section).  But before that, it is illustrative to see a physical replica (absent the tick marks which will be later examined experimentally along with a non-transparent immersion bowl) below:

 

How the hole patterns allow for interesting resolution read-points (6 unique angles corresponding to 12 faces for a dodecahedron) of the displacement profile will be later discussed. In order to understand the ideal measurement strategy for utilization of the aperture patterns, the mathematics of uniqueness should be considered, and the indented rings surrounding the fluid apertures will be further detailed in the design nuances section. 

 

Mathematical Description of Uniqueness

A revisit of the discussion for the need of uniqueness, based on mathematical development, is warranted.  In a simplified 2 dimensional case, a shape can be approximated by a grid of pixels, with each pixel value having either a 1 (mass is present) or a 0 (no mass belonging to the object), as shown (see figure 6):
hypothesis_6

The original and reconstructed patterns in figure 6 consist of a 6x6 pixel grid of values that indicate whether mass is present or absent (lack of mass is represented as black pixels).  Even though both the vertical sum (i.e. vertical fluid volume displacement) and the horizontal sum of the two patterns match, the patterns themselves do not match.  Therefore, a fluid-displacement measurement performed at 2 angles is not sufficient for providing an accurate description of the shape of the original pattern.  The question becomes, how many angles of measurement should be performed to provide a unique description of the original pattern?  How do the potential resolution read-points (12 sides, 6 unique angles) provide the measurement utility and mathematical description of the object within (see figure 6.5)?figure 6p5
For a linear system of equations, the number of equations supplied must match the number of unknowns for a guaranteed solution.  However, the variable values, in this case, are constrained to either be a 0 or a 1, so the number of equations can be reduced, with the amount of the reduction dependent upon the shape of the test object to be measured.  In figure 7, there are 36 unknowns, and 12 displacement equations (6 horizontal plus 6 vertical).  Although there is not enough equations to provide a full solution and a matching reconstructed pattern, the degree of pattern matching accuracy is highly dependent upon the original pattern shape. To illustrate, a computer is used to solve the following equation using a linear combinatorial binary solver:

hypothesis_equation_1

The element A is a square matrix identifying the pixel variables contained within the relevant displacement "slice", x is the presence or absence of matter inside the pixel, and b is the volume displacement of the slice.  For the original pattern of figure 6, the variable assignment to the matrix equation is shown (see figure 7):

 hypothesis_7

As illustrated previously (see figure 6), despite the satisfactory solution of the matrix equation, the 2 angles of volume displacement measurement were deemed insufficient to create a matching reconstructed pattern. To study the dependence of pattern reconstruction accuracy on the original pattern shape, the equation in figure 7 is solved for a number of example cases, as shown (see patters 1-4):

pattern_1_to_4

Based on the fact that the number of equations is not sufficient to solve for the number of unknowns (assuming 2 angles of displacement measurements), the reconstruction accuracy is not 100% for all patterns. The degree in reconstruction error rate is proportional to the sparseness of the original pattern. This result can be understood by examining the number of cases where a single displacement measurement eliminates the uncertainty of the existence of matter within the entire slice, as illustrated (see figure 8):

row_cancellations

Evident from pattern 3, as highlighted, there is one vertical column where a displacement sum is 0.  In that case, it is known that all 6 pixels in that column must not contain matter. The result is a lower error probability (22%) due to an effectively reduced ratio of unknowns to equations.  In pattern 4, the situation is  even more dramatic, where there are 4 displacement slices that contain no matter, effectively cancelling the uncertainty in 24 pixels.  Thus, it is clear why the shape of the original pattern affects the accuracy of the combinatorial reconstruction.

3D Mathematical Simulation Program

Now that the foundation for the idea of volume sum reconstruction has been established, extension of the concepts into 3 dimensions can begin.  While computers were certainly not available at the time of the design of the Roman dodecahedra, simulation allows the opportunity to study the design choices made.  A program to render a 3 dimensional shape based upon the incremental volume displacement "slices" was written (see figure 8):

hypothesis_8 

Similar to the 2 dimensional case, the A matrix is populated with variable unknowns.  However, an unknown in the A matrix represents a voxel (a cubicle version of a pixel that contains spatial volume).  As a result, a straight volumetric "slice" through a grid of voxels may involve only a partial volume intercept of the voxel.  The solution to the matrix equation (A*x = b), in practice, and even computationally, will result in a value for x that is not limited to a value of zero or a one.  There are 2 reasons for this:

      1.  Solver routines are trying to best optimize the solution within a limited computational budget.

      2.  The reconstruction accuracy is not 100% for all patterns.  The simulation is taking into account 6 angles of fluid immersion, each different "look" to the object, given the many permutations of angles which could be used to generate a solution to x, is trying to best satisfy each visual perspective.  Certain angles are superior for imaging certain regions of the object, based on reasons similar to the 2-dimensional case (absence of mass within a certain slice at a specific angle can reduce the number of unknowns), and the resulting solution for x is an attempt to satisfy all angles of measurement.

      3.  The solver has no motivation to provide physically accurate solutions.  For example, it may prefer to add floating mass particles of dust suspended in the air to satisfy a displacement sum (though that would show up as a Dirac delta impulse in the 1d signature and could be correlated out by setting thresholds that look for the impulse to occur at specific XYZ locations at other rotated angles).

It is necessary to slightly relax the constraints on the solution for x. In the 2 dimensional case, the solution was constrained to be quantized (0 or 1), but it will now be relaxed (from -0.5 to 1.5 typically). After solution, the routine assigns a lower and upper threshold (based upon a statistical engine) to determine whether a voxel contains displacement mass. More details on computational methods will be later discussed. 

 

Displacement Signatures of Ellipsoid and Example Reconstruction

It was previously shown how a sphere's displacement profile was measured through incremental immersion.  A sphere is a special case of an ellipsoid where a = b = c in the below equation for an ellipsoid:

equation of an ellipsoid

The sphere only required a single angle of measurements to find its displacement characteristic, because of symmetry, and thus no immersion cage was required (although strictly speaking the sphere needed to be checked that the signature repeats at various orientations).  For a tri-axial ellipsoid, which only has rotational symmetry along its axis, or more complicated shapes, multiple angles of measurement are required to discern shape, and as a result an immersion cage is needed (dodecahedron or icosahedron are logical candidates).  Intuitively, based upon the results of the experiment with the sphere, the consideration of the problem of measurement of uniqueness, and the limited degrees of freedom of a simple ellipse, one would postulate that a measurement cage, with various hole aperture points to record the water level required to transcend from one tick mark to the next, a measurement tank, and a DUT holder are all that is required to describe and experimentally measure the parameters of an ellipse (illustration below):

 It is instructive to see how a dodecahedron immersion measurement can mathematically reconstruct an ellipsoid in order to demonstrate potential utility of the device.  The simulation program was utilized to construct an ellipse for displacement "slicing."  The device under test is oriented in the vertices within the boundaries of the dodecahedron as illustrated below:

 

Because of the 3 parameter symmetry of the ellipsoid above, it would be expected that there would be unique displacement profiles resulting from the 6 unique angles of measurement of the dodecahedron.  Illustration below shows the corresponding displacement profiles from the simulation program (noise is due to voxel intercept grid error):

 

The displacement profiles are then injected into the A matrix, and solved, giving the below reconstructed image:

After solving the linear equation for x, rounding was performed as shown above.  The 2nd solver iteration was subsequently performed by removing elements of the matrix (and their corresponding displacement sums) that exceeded a statistical threshold, and the resulting 2nd iteration reconstruction of the ellipse had exactly 0 voxel errors within the 16x16x16 reconstruction, below:

 

 

Clearly, from the results of the computer experiment, a simple ellipse is unique described by the displacement profiles that can be obtained within the angular resolution of a dodecahedron cage. Before going further into the theory and mathematical details of volume displacement 3d reconstruction, it is instructive to review the implications of these results to the design of the historical artifact.

Displacement Signatures of Ellipsoid and Example Reconstruction

Various specimen of Roman dodecahedra, including one icosahedron that was found, are shown below:

 


At first order, the variation and construction of specimen generally resemble what one would expect if creating an immersion cage for 3d volume incremental recording; the design of the cage should be optimized for the device under test in order to achieve the right combination of fill factor and imaging resolution for the target application--a variety of cages would be necessary to deal with a variety of imaging targets.

For example, if one was to measure the shape of sling ammo (i.e. ellipsoidal) one would prefer specimen #4, because the largest possible ellipsoid could be placed within the large entrance hole.  The number of resolution points, and angles, need not be large, based upon the previous uniqueness mathematical treatment.

Specimens 1 through 3 would be of preferable design for the case of an elongated target, because for a thin target, a high fill factor cannot be achieved about a symmetric rotation within a measurement tank.  As a result, one would design the low-fill factor cage to have a high degree of bulk in its thickness and feet, such that a large fluid rise would be modulated by a small displacement mass.  In imaging, this gain ratio is called "conversion gain", and reduces the readout noise of an imaging system (improves signal to noise ratio).  It reduces the effect of fluid displacement measurement error on the accuracy of the dataset.  This concept will be further discussed.

Specimen 5 is an icosahedron.  The choice to use a 20-sided platonic device, as opposed to a dodecahedron, can be justified from the table in next section.  Despite lower fill factor, the increased angular resolution, and the high number of water level read-point holes, can be an optimal choice for certain targets.

Second order analysis is required to understand the nuances of each variant, such as the shape of the "feet", the concavity of the specimen #5, and the size, depth, and width of the indented rings on the outer surface.  It is by focusing on the nuances that one can determine the purpose of the Roman dodecahedra to high degree of certainty.

At this point, it is very instructive to step away from the claims made here regarding the 3 dimensional Roman dodecahedron, due to the complexity of understanding the ideal geometries for imaging sensitivity. By studying a modern 2 dimensional pixel design, the appropriate language of constraints can be established (next section).

Design constraints and solution for a 2-dimensional modern imager to serve as a modern analogue of the geometrical considerations of the design of the Roman dodecahedron artifact

The CMOS image sensor pixel is responsible for collecting photo-signal in modern day cell phones. A "4T" CMOS imaging sensor consists of a photo-sensitive region, and electronics to read out the signal gathered by the light-sensitive region. The pixels (individual dots in an image detailing the intensity of light and the color) making up the image taken from a modern camera, is likely to be using such an architecture. One benefit of a "4T" device is that correlated double-sampling allows for a high cancellation in the effect of geometric and physical noise, and variation from pixel-to-pixel of the same:

 

To understand the power of CDS, one might consider that modern CMOS pixels are trending towards the wavelength of light, with production pitches in the 1.1 micron range, yet image quality continues to improve, relative to a given sensor area.

Constraint #1: Utilize the error compensation of CDS

The fill-factor of a CMOS image sensor is the ratio of the light sensitive area, relative to the entire area of the pixel.  More generally, it can be thought of as the efficient use of space.  In the trunk of an automobile, if the spare tire encompasses most of the room that could be available luggage, then fill-factor of the trunk is small.

The higher the fill factor of a pixel, the larger the signal generated by photons to which can be stored, increasing the maximum signal-to-noise ratio.  Fill factor can be improved in three ways:

Constraint #2: Reduce the area of readout electronics

Constraint #3: Minimize the remaining isolation space

The shape of the fill-factor optimization will impact the manufacturability and ultimate cost of the sensor.  If a target fill-factor is achieved through a very non-symmetric (relative to the center of the signal-sensitive region) manner, then cost goes up, and signal-to-noise ratio goes down, for two reasons.  First, the fill-factor would not, in that case, surround the center of the light-sensitive region, not allowing full capture of wanted signal.  Second, the length of the boundary region between light-sensitive and non-sensitive regions increase, and edge regions contribute to unwanted signal (through boundary interface stress effects) and manufacturing consistency (hot spots during thermal bake process) problems, giving constraint #4:

Constraint #4: Increase the fill-factor with symmetry in mind

The maximum symmetric area, relative to signal reception region center, in 2 dimensions, is a circle.  However, that does not mean that additional benefit can't result from adding additional fill-factor by adding radially non-uniform regions (such as replacing the circle with an octagon).  Instead, the idea is that the benefit of any additional fill factor goes down the further added photo-receptive area is (power receiving area) from the optical center of the design.

The same idea, but in reverse, is that the readout electronics at the farthest region from the optical center, for a given area, has a quality that is represented by its symmetry of distance around its own center.  A non-uniform area around that far center point implies a stressful shape that is closer to the optical center of the pixel, which violates the following principle:

Constraint #5: Increase the symmetry of readout electronics around its own center point

Regions where thermal gradients are large, such as corners, should be placed away from interface of the photosensitive region.  Power supply voltage, and stressful contacts, should be placed as far away as possible.  And isolation regions (where crystalline silicon has been etched) should be minimized.

Constraint #6: Sharp corners, interface regions, hot spots, damaged regions, and power supply voltages should be minimized and placed as far as possible from photosensitive region

The circle must be placed within a square for the highest possible symmetric fill-factor of a given array format (in order for the circles radii to touch adjacent circles).  Determining the fill factor of a circle inscribed within a square was once considered a daunting challenge, but per Archimedes' [287 to 212 BC] "squaring the circle approach", it was found to be as follows:


 

As an interesting aside, there are a number of interesting ways to fill in a cylinder using triangles.  A technique the author developed during graduate study, is a binary tree-branching fill-in that can grow the mesh, ring by ring, with a decision to branch or not-to-branch based on error convergence as the number of cells grow (in this case error convergence was part of a classical numerical electromagnetics consideration where a cylinder illuminated with radiation scatters electric field from every point within such that the boundary conditions at the surface are maintained.  The ring-by-ring binary decision approach allows the decision of cell density to be made around a chosen number of fundamental and cross-section-able angles, with N not necessarily needing to grow exponentially large to maintain fundamental and cross-sectional lines of symmetry, and maintaining the ability to grow the number of cells adaptively based on the trend of convergence to a final solution for electric field and phase angle:

  

In 3 dimensions, a sphere can be placed within a cylinder (per Archimedes tomb). Per Archimedes, the fill-factor of a sphere inscribed within a cylinder is 2/3 (it will be interesting to compare to the fill-factor of Archimedes 9th solid, or the dodecahedron artifact, rotated within a sphere):

   

Now that some specific geometrical concepts of fill-in have been discussed, we might return to the 2-dimensional CMOS image sensor design modern day analogue.  One might consider the tradeoffs between 3 "layout" alternatives (layout is a floor-plan on dividing the light sensitive region area with the readout electronics/isolation region):

It can be recognized that the spherical photodiode has the largest symmetric region about the center of the light-sensitive region.  The octagonal design has a slightly larger fill-factor, with very little impact to the symmetry of the photodiode, but a large improvement in the symmetry of the readout electronics (area in green).  The rectangular design, which was an industry standard conventional design back in 2005 in the CMOS imaging field, has both a non-symmetric photo-receptive region and a very poor symmetry, and quite a complicated shaped region for the readout electronics.

What is striking about the octagonal design is how effectively it creates the combination of a highly symmetric photodiode region, and readout electronics--just a slight change in the photodiode shape caused a large improvement in the shape of the readout electronics/isolation region (see below):

 

As highlighted above, the readout electronics (shown in green) are much closer in shape to a circle (i.e. symmetric around their own center-point) for the octagonal design than for the circular photodiode region design.  It is quite evident that an octagon is a reasonable compromise (very little room for argument that it is best 2 dimensional compromise) between a perfectly circular photodiode, and a perfectly circular readout electronic/isolation region.  There is very little deviation one can make from this ideal octagonal design and provide a reasonable tradeoff in light of previously listed CMOS image sensor pixel design constraints (c1 through c6).  Although a table comparison is somewhat subjective, and dependent upon the design rules of the imager, the overall conclusion is certainly not:

 

The tabulation above clearly implies that there is a demonstrably optimal image sensor photodiode design, an optimization that, considering the overall tradeoffs, is not a simple matter of subjective opinion. In 2006, the author designed an experimental pixel design (U.S. patent issued and additional details presented in ISSCC 2011) that captured the elements of these constraints:

The performance of the design provided record performance with respect to read-out noise, dark current generation, fill factor, photo-response non-uniformity, and manufacturing variation (signal response and threshold variation).  It is illustrative to consider the details.  Unwanted energy, from the power supply, is as far as possible (given the design rules) from the periphery of the light-sensitive region.  Similarly, the distance between the core of the light sensitive region, and the periphery edges, were maximized to avoid fill-factor loss.  The interface region is minimal relative to the fill-factor, and the readout electronics are highly symmetric.  The transistors effective width can't vary, because of the self enclosing structure.  By making both the photodiode region and the readout electronics octagonal, symmetry is achieved in regards to constraints (c4 and c5).  An image that was captured utilizing the above design is shown:

At this point, the reader might suspect that the author is proposing that the Roman dodecahedra artifacts are a 3-dimensional analogue of the pixel design with octagonal photodiode layout, and optimized for c1-c7.  This is indeed the case. Although it took time for the CMOS industry to recognize the octagonal advantage, the author suggests that the likes of Archimedes may have rapidly conjured up the ideal aperture hole pattern and platonic structure (Archimedes 9th solid) that would be applicable for field use, extending his volume displacement principle to angular and linear increments. He may have further generalized an instrument (the Gimbal) that could arbitrarily extend the angular support of a test object.

In 2 dimensions, it was found that an 8-sided structure provides optimal photodiode and readout area symmetry around a signal-gathering core. Extending from 2 to 3 dimensions correspondingly increases the number of platonic sides from 8 to 12 in order to maximize the fill-factor of the platonic shape while minimizing the edge length, relative to the size of the signal aperture (proportional to the face area).  There is no other 3d platonic structure that has a higher fill-factor (volume that can be inscribed within a sphere of rotation) relative to the entrance hole (i.e. face area), while minimizing periphery length. This is not immediately obvious and difficult to derive.  A comparison of several possible structures is shown:

 

Intuitively, one might suspect that both the dodecahedron and the icosahedron are reasonable choices of structure for the given constraints, with their subjective scores of 4.6 and 4.0 respectively. The dodecahedron offers increased fill factor, aperture size, and minimizes the periphery length, and is simple to manufacture. The icosahedron has some challenges, and SNR will be inferior (caution that statement depending on imaging target ROI) but angular resolution is benefited.  The Romans mass produced the dodecahedron version and experimented with at least one icosahedron variant (similar to the author who considered but ultimately dismissed icosahedron prototype, unaware of ancient historical artifact).

One can conclude this separate 2 dimensional modern analogue by recognizing that an imaging system, whether 2 dimensional electrical circuitry, or an ancient 3d geometric measuring device, face similar measurement accuracy and manufacturing challenges, and as a result, an optimal design can be reasonably derived. The manufacturing constraints will be touched upon in greater detail.  The fact that even the 2 dimensional solution is not obvious can be evidenced by US patent US2008/0258187 A1, the result which came from years of study in a competitive multi-billion dollar industry, where layout and pixel design engineers work around the clock for any improvement in performance.  Clicking on the image below pops up the patent pdf of the octagon/square solution for 2 dimensional analogue: 

 

 Clicking on above image reveals 2-dimensional octagonal/square solution for c1-c7

 

One could argue that a 3-dimensional design, based not only on the constraints of c1-c7 but carrying the challenge of low light performance, manufacturability, fluid level settling, air-bubble traps, and practical laboratory utility, would be beyond the engineering and mathematical toolsets of anybody in antiquity.  However, there is one man who may have conceivably fit the profile to have secretly developed an elegant solution to the problem, a solution so powerful that the design would be painstakingly manufactured for many hundreds of years.

To explain why the purpose of the Roman dodecahedra's, under proposed hypothesis, has not been revealed, one must consider that a simple and elegant solution to a specific problem may be undecipherable without knowing the specific problem to be solved.  Sometimes, complicated solutions arise as the answer to simple problems, and in other cases, simple and elegant solutions result from quite complex lists of constraints.  In the latter case, there may have been only one inventor in the world with the mandate, purpose, and background to provide the unique solution, and elegant solutions can be readily mistaken as artwork.  But when presented with the specific problem to be solved, those skilled in the art may develop thousands of solutions that all fail to offer practical advantage over the elegant and timeless solution.  Without knowing the specific problem that is being addressed by the elegant solution, a wide population of attempts may be made to decipher the instruments purpose, but the challenge is daunting.  First, the population is being asked to solve a riddle for which the appropriate background experience is extremely rare.  Second, the population is challenged to decipher an instrument for which a simple, unique and elegant solution resulted from a complex problem.

Consider the case of the discovery of a wheel on an axle.  A circle that can be rotated could serve as a solution to many simple or complex problems.  Only nuances in the design would reveal without ambiguity the potentially complex set of constraints for which the solution fell out.  An example would be a metal grooved wheel (low rolling friction) that solves the problem of locomotive travel on a rail-track.  Without having been presented with the problem of a need for a large capacity and long distance transport, the purpose of a grooved metal wheel would be anybody's guess.  

2nd Order Analysis of "Sling Ammo Variant"

To understand the nuances of specimen #4, a blank replica (absent the indented rings, or precisely shaped feet) was manufactured based upon CAD analysis by CAD engineer Josh Turk.  Specimen #4 will be referred to as the "sling ammo variant" for reasons to be explained:

 

To understand the nuances of the above specimen, a blank replica (without the indented rings, or precisely shaped feet) was manufactured, and then a handheld motorized grinder was utilized to carve the shape of feet or depth and width of the indentation rings, and study the design impact of each change.  Field testing was performed to understand the design impact on practical operation and measurement.

Shape/size of feet and tick mark groove depth/width

To study how the nuances of the feet and tick mark design affect the accuracy of measurement, one must consider the environment of operation. For example, the physical measurement with the historical roman dodecahedron artifact would be performed without a glass immersion bowl. For the previous illustrative case of the sphere within a clear tank, the eye can be made level with the surface of the immersion fluid, and the fluid height can be directly read off of the ruler:

However, for the historical case, transparent containers were in their infancy and could hardly be referred to as such.  Consequently, an opaque bowl was likely used, and the measurements were made on the top half of the dodecahedron only because the lower half of the dodecahedron would be obscured.  The fluid level would have been read directly off of the dodecahedron aperture holes and indentation rings (i.e. tick marks).  In addition, because there is only a limited number of resolution points, it would seem likely that the users would record the amount of fluid required to transition to adjacent tick mark rings, rather than applying a specific amount of fluid and recording the change in height.  For example, they could utilize a fluid level dropper, and record the number of drops required for the fluid level to fill from one ring to the next.  Performing this operation on all 12 faces (only recording the top half of the dodecahedron due to line-of-sight issues of opaque immersion tank) would be necessary.  The resulting measurement table would be a 12 column grid, with the number of rows equal to the number of tick mark rings and aperture hole entrances (i.e. resolution points).

A second consideration of the line-of-sight issue is in regards to the ability to visually discern the fluid level location relative to the tick mark.  For the transparent sphere and ruler immersion setup above left, fluid level can be easily determined by bringing the eye level at same height as the fluid level.  For the opaque bowl, it is difficult to see the fluid level relative to the entrance hole when looking at a downward angle.  The problem was considered quite severe during field trials, studied extensively, and a practical solution was found that gave further credence to the overall theory.

Challenge of measuring fluid level at downward view angle

To study the accuracy of the fluid level measurement, a weight scale was placed underneath the tank, to eliminate any uncertainty in the amount of fluid applied. The noise in the visual fluid level measurement could be isolated by recording how much mass was required to be added to the tank to visually reach from one tick mark location to the next. If this was not highly repeatable under realistic use conditions, then the visual error itself would limit the practicality of the historical device.

After significant trial and error, it was recognized that the reflectance of the tick marks, and the size and shape of the feet had a direct impact on measurement accuracy. To improve upon visual contrast, an attempt was made by painting the face of the dodecahedron black, the feet with a highly reflective gold paint, and the tick marks were abraded to various depths to study the impact on visibility:

 

The final test structure is shown below. It includes a challenging ellipsoidal shape that is quite thin (i.e. low fill factor) relative to its length and width:

A laser beam was utilized to study the impact of the tick mark ring width/depth variations and return of the signal to the human eye.  It became immediately evident that the feet themselves were the most effective at channeling incident light to the tick marks and then to the eye.  This was quite a surprise, and yielded significant insight to the design of the dodecahedron and to the environment under which it may have been used.  Simple ray tracing illustrates this important result:

 

Above, it can be seen how a sharp return to the eye can be achieved based upon the fact that incoming light can be reflected across the face of the dodecahedron and highlight the groove.  A simple alternative to illuminating the feet with a laser, is to use the reciprocity theorem.  In electromagnetics, it is often convenient to replace the source and receiver.  In this case, a hemispherical LED can be placed at the aperture surface, and light shining tangent to the dodecahedron surface bounces off the feet and towards the user:

Naturally, the shape of the feet impact the amount of light returned to the eye, and this is fertile ground for evaluating strength of hypothesis. A spherical "foot" may not be optimal for semi-collimated light.  And the feet of specimen #4 more resemble a "cotton swab" than a round ball--ray tracing may explain the reason as illustrated below:

Despite optimizing feet shape and tick mark width/depth, field experiments continued to show visual problems for the user when utilizing a historic setup with opaque bowl.  Then a breakthrough occurred.  Instead of using laboratory lighting, a candle was utilized inside a pot with holes, and by holding the pot in a specific position intended to provide energy to the bottom feet of the dodecahedron, and the water level clearly lit up "like a Christmas tree."

 

Such a result can be expected from Snell's law.  If the candle is positioned such that the bottom foot reflects significant light at the 48.6 degree critical angle of water to air, then scattered light will be visible from the water line.  The key, it seems, to getting a visual water level is utilizing a point source (i.e. candle) within an aperture type holder (a holed pot or perhaps even another dodecahedron if available) and to rotate the light source until illumination of the bottom submerged foot of the dodecahedron.  For water levels exceeding the 3 "feet" of the pentagon, then either of the pair of submerged feet can be utilized to reflect light from the feet to the water-air interface.  By having many holes available in the pot, a simple rotation of an elevated illumination pot is all that is necessary to achieve such a result.  ADD VIDEO LINK DEMO

2nd Order Analysis of "Icosahedron Variant"

After examining the nuances of the sling ammo variant, the task of deciphering the icosahedron variant design is less daunting. For convenience, the icosahedron variant is illustrated again below:

 

It can be observed that the DUT placement hole is located on the side that has smaller feet.  Consequently, there are reasons to suspect that the sides with smaller feet are not utilized in measurement (i.e. face is never down during an immersion).  First, the visibility of the tick marks was previously found to be a function of the light power reflecting from underneath the water to the surface at the critical angle.  For a high resolution measurement device, such as the icosahedron, fluid level visibility would be high desirable for speed of measurement--the sides with small feet would be more difficult to use.  One might ask why there are any feet at all on those sides, but it can be explained from thermodynamic constraints one faces with the lost-wax method (discussed later).  Thus, they may be considered "dummy feet" that are necessary for thermal symmetry during the manufacturing process to avoid warping of the cage.  Another reason the associated faces may not be utilized can be understood from the small entrance hole.  For the dodecahedron, it is a simple matter to snap the DUT (Device Under Test) holder into the corner vertices, or rotate a screw holder into place (both methods have worked well in tests).  Due to the concave face structure of the icosahedron specimen, and limited entrance hole size, it would be more difficult to negotiate a target into place.  For that reason, one might propose that the DUT holder extends beyond the DUT entrance hole and may also serve as a rotation handle, such as in figure below:

 

One might expect a high number measurements to be required to uniquely describe a device under test, and a higher number of measurements per face.  This could be achieved by modifying the regular measurement sequence.  Instead of recording the amount of fluid required to transcend adjacent tick marks (and repeating until all tick marks had been transcended), angular change could be done between tick marks utilizing buttresses.  In order to fully understand the measurement sequence, it would be necessary to know the device that is being characterized.  A convex device may further limit the faces utilized, because of potential air bubble traps, or may require a rotation between measurements to fill an empty cavity before displacement measurement can occur.  Let us consider the concept of cavity traps that require a rotation.  An example imaging target is shown below:

 

Illustrated above is a positive error in the displacement measurement from fluid level 1 to 2. However, a subsequent measurement (below), from 3 to 4, would have a negative displacement error due to the same cavity:

Displacement measures of both figures contain the same error magnitude of opposing polarities, positive in the first, and negative in the subsequent measurement.  One way to avoid the error in the 2nd measurement would be to do a rotation that fills the cavity before water line 3 above is exceeded, as illustrated below:

The icosahedron design needn't be justified on the sole premise of avoiding air traps for a concave target. A more direct advantage of this variant comes from the fact that there are only 3 feet per face. Because the face is of smaller area than the 5 sided pentagon of a dodecahedron (relative to a given bowl size), a smaller buttress is required to achieve a higher angular rotation of measurement, partially mitigating the disadvantages in angular swing that would stem from having certain faces unusable if the DUT holder extends outside the entrance hole. Instead of remaining at one angle and measuring subsequent pours, a hybrid approach could be utilized where after each pour, the displacement is measured, and an additional data point is achieved by rotating as shown above.  Such an approach would certainly increase the efficiency in which one could record a descriptive dataset along a limited cone angle for the target under test.

Justification for concave faces

A striking feature of the icosahedron variant is the concave shaped faces.  Considering that the size and shape of the feet was previously seen to impact the low-light performance of the dodecahedron, one might propose that the concave shape design of the icosahedron faces were of optical considerations.  Simple ray tracing would suggest that a higher proportion of incident power would be directed towards the center of the trace:

There is secondary indication that the designer of the icosahedron variant placed a greater priority on the center hole of the face. Examining the face, one sees that the center resolution point has a large number of slight indentations surrounding the center hole below):

By comparison, the dodecahedron variant has a large entrance hole, and as a result, the volume consumed by the imaging target would extend to the edges of the inner cage.  Resolution points across the entire interior of the cage would be of great interest, and the tick mark strategy would reflect that.  For a given ellipse ratio (the length twice as great as the width), a comparison of the span of the regions of interest for an ellipse that could fit within the entrance hole of each respective cage is highlighted as follows:

 

Although there may be reason to prefer more reflected power incident on the center of the face, the justification based upon region of interest seems insufficient.  Perhaps a stronger motivation is from the premise that the DUT holder renders half of the faces unusable. Since an extra space in the bowl around the cage would allow angular play, it would be possible to see the bottom half of the icosahedron during immersion measurement. This is of great practical benefit for the limited rotational case. Utilizing a concave face would thus further increase visibility to the bottom half of the tick marks, allowing line-of-sight:

The ability to increase viewing angle, possibly allowing twice as many faces to be visible during measurement, also enhanced by the large feet of the icosahedron variant, and thereby negating the limitation imposed by a DUT holder that extends from the entrance hole, would be the strong justification required to explain the design choice made to use concave faces for icosahedron variant. The dodecahedron variant, however, does not require the ability to see the bottom faces, since the self contained DUT holder does not prevent flipping any of the faces to the bottom of the bowl, and all faces can therefore be utilized. 

Performance Evaluation of the Roman Dodecahedron: 

The device must have a practical repeatability, signal-to-noise ratio, and resolution, in order to be a useful 3D quality control and shape recording device.  Let us first consider repeatability.

Repeatability of DUT placement and human error in judged fluid level

It is required that devices to be test can be lodged into the corner vertices consistently in order for displacement measurements to be repeatable.  The illustration below highlights the navigation process into secure placement:

As seen in steps 1-4, the DUT with screw holder can be navigated into the corner vertices, through the large entrance hole, with one hand, such that the DUT is held into place.  A slight rotation of the DUT (about 1/16th turn) secures the DUT tightly.  As a side note, it is interesting to think about Archimedes contribution to the mathematics of the problem of negotiating a linear object around a corner (the same problem furniture movers face when moving a long object such as a couch into a small room), and that he might have likely chosen a screw holder (similar to above based upon his work with screw machines).

It might be reasonable to assume the use of a screw to increase tensile compression for securing DUT placement, similar to that shown, considering Archimedes knowledge with screws and that rotation to achieve DUT fit within vertice complements the broader concept of dodecahedron rotation within a bowl.  One can imagine him calculating what the fill factor of the cage (his 9th Archimedean solid) within a sphere, which also results in his famous 2/3 that he is said to have requested be placed upon his tomb after his death.  Alternatively, a slightly flexible DUT holder, such as a wooden piece, was found to secure a DUT into position with high repeatability.   The historical sling ammo variant design has a large entrance hole that seems to perfectly match the requirement of negotiating a 2/1 elliptical object across the resolution pattern span of us; working with the device in the field, gives an extra degree of confidence in the theory that is difficult to translate onto paper.

To test repeatability, the dodecahedron was placed on a precision weight scale and fluid level was added until a tick mark level was visibly reached.  The amount of fluid required to reach this tick mark level was precisely recorded with a weight scale to rule out human error and in order to isolate DUT placement repeatability--other user errors will be considered later.  In order to record the angular position of the DUT within the dodecahedron, eliminating it as a source of error, a lamp was used to project the shadow of the dodecahedron structure, and the sling ammo, into a whiteboard, as illustrated in both figures below: 

Please see HYPOTHESIS, PART 2, continued at the following link: http://www.romansystemsengineering.com/hypothesis2.html<