Hypothesis (part 2)

CONTINUED FROM PART 1 (previous part is at: http://www.romansystemsengineering.com/hypothesis.html )...

 

Once the DUT rotation within the dodecahedron was ensured, the dodecahedron was placed in a tight-fitting immersion bowl and the amount of fluid to transcend two chosen tick mark rings were recorded, as labeled in illustration below:

The process of placing the ammo into the structure, immersing, and recording, was repeated 3 times, with corresponding images below:

 

The summary of the measurement results are tabulated below:

The summary table above gives an incredible insight into the utility of the Roman dodecahedron artifact.  It can be interpreted that a test object that has an incremental displacement volume, dV_DUT, within a cylinder of incremental displacement volume dV_cyl, can be measured with a signal-to-noise ratio of 32.  It states that the small region of the sling ammo only encompasses 16% of the volume of the cylinder for which the fluid level transcends between tick marks, yet a high signal to noise ratio can be achieved.

The weight scale was utilized as a way to characterize the repeatability of the DUT placement, and the human error in the visual discerning of fluid level.  The sling ammo replica measured an incremental volume displacement of 16 grams for the region of interest between tick mark rings 1 and 2, with signal-to-noise ratio of 32.  The fill factor of the sling ammo within the tank can be calculated as the ratio of displacement to the volume required to translate the fluid level across the measurement rings.  It is important to quantify the SNR measured to the fill-factor of the displacement slice, as the SNR would be expected to trend proportionally to the fill-factor.

Note that the empty dodecahedron was measured for the purpose of CDS (double-correlated sampling).  By measuring the fluid level required to transcend tick marks 1 and 2, while the cage was empty, any deviations in the shape of the structure will have no effect on the accuracy of the displacement measurement.  The extension of this SNR result in the practical application of the Roman dodecahedron as a 3D shape recording device that can be utilized to record and compare two 3d shapes for likeness can be done.  At 16% fill factor, its incremental displacement can be measured with 1/32 = 3.1% error.  Assuming that SNR is a linear function of fill factor, one may estimate that an object that was manufactured with 6.2% lower incremental volume than desired can be flagged with 2 sigma of confidence.  And given that this form of 3 dimensional spot checking can be done at 6 unique angles, and at intervals across the entire span of the object, one would conclude that the Roman dodecahedron artifact can effectively serve the proposed purpose of allowing a measurement table specifying the shape of the sling ammo with practical utility.  Thus, the practical utility of the Roman dodecahedron has been experimentally proven.

The reader, at this point, may be concerned with the accuracy of the supplied fluid amount that was recorded to transcend the tick marks.  The users of the Roman dodecahedron needed to do more than place the DUT in repeatable fashion and discern the fluid level accurately.  They also needed to be able to record the volume amount of fluid required to transcend tick mark rings.  That gives rise to a question of what their likely fluid level delivery scheme was.  For example, in modern times, a pump might be utilized to deliver a quantity of fluid with 1 mL accuracy.  Alternatively, a weight scale can record the mass added, perhaps with 1 part per 10 million in accuracy.

In antiquity, there was certainly the ability to use a fluid dropper, and although there is error with use of a fluid dropper, it is extremely small when a large number of drops are supplied.  A flask, or a water clock, or alternatively a balance or perhaps a miniaturized version of Archimedes screw could be envisioned.  In the case of a balance, the chance in mass when the fluid level transcended a tick mark ring could be recorded, similar to above.

Need for Shape Measuring Device to Improve Battle Effectiveness

Ancient writings commonly emphasize a large advantage of range of the sling weapon compared to the bow, being able to be slung in excess of 400 meters.  It was a powerful weapon capable of defeating armor, not relying on penetration, but through sheer percussive shock.  Sling bullets are common finds throughout antiquity and vary vastly in shape and size, appearing to be optimized for a number of constraints, including aerodynamic flight and how the ammo sits within the cradle during release.

Hill forts and protected military sights were found that appeared to be designed around the use of the sling, with the number of concentric ramparts optimized relative to the terrain slope and width, maximizing the efficiency of hailstorm of sling bullets on advancing troops.  At Maiden Castle, Dorset, a cache of 40,000 sling bullets was found.

To understand the significance of the sling as a weapon of war, one may refer to the treatise of Roman military tactics, "De Re Militari" (Latin "Concerning Military Matters"), written by Publius Flavius Vegetius Renatus.  His work is the only acient manual of Roman military institutions to have survived intact.  He wrote:

"It is universally known the ancients employed slingers in all of their engagements."

The sling is mentioned by many Greek authors, along with Homer.  In the retreat of the Ten Thousand, 401 BC, the Greeks were decimated by the slingers in the army of Artaxerxes II of Persia.  However, the Greeks responded later with lead sling ammo.  According to Xenophon, their lead ammo struck targets at twice the range as the Persian slingers utilizing stones. 

Sling ammo projectiles vary in size from pebbles to fist-sized.  Projectiles were baked from clay, allowing high consistency of size and shape to aid long range accuracy.  The most advanced ammunition was cast from lead and was widely utilized by the Greeks and Romans.  Leaden sling-bullets vary significantly in shape and size.  Besides spherical, there is ellipsoidal, and biconical (resembles a flattened American football, or an almond) variations.

Almond shaped leaden sling-bullets were typically about 1 3/8 inches long and about 3/4 an inch wide.  Many examples have been found, including about 80 sling-bullets from the siege of Perugia from 41 BC.  Due to the importance of the sling weapon in defending a siege, the effect of sling ammo design on the outcome of war, the resources devoted to sling ammo manufacture, and the variation of designs across Greek and Roman antiquity, one may surmise that there was a need for a practical method to record and tabulate the 3 dimensional shape of sling ammo to enhance battlefield effectiveness.

Time Interception of Technology with Practical Need

The technique to measure a volume of an irregular shape was invented in Archimedes (287 BC - 212 BC) time. Archimedes demonstrated that the volume of the solid is equivalent to the corresponding fluid level rise. In addition, he realized that an incremental immersion of an object would displace a corresponding sub-volume. A natural question would be whether Archimedes considered various angles of immersion, and how incremental immersions recorded at various angles would identify the shape of the object measured.  This inventive leap between incremental immersion, and shape, would require at least a rudimentary understanding of combinatorics.  In the Ostomachion, Archimedes considers a tiling puzzle.  Combinatorics often involves finding the number of ways a given problem can be solved, subject to well-defined constraints.  In the case of 3D volume displacement recording, the goal in designing the immersion instrument and procedure is to essentially maximize the uniqueness of the dataset that can be achieved through practical use.

If it had occurred to Archimedes that a finite number of incremental immersion increments, taken at a small number of angles, can effectively record the 3 dimensional shape of a simple object (ellipsoid etc.), it may have occurred to him to use a platonic immersion cage (he developed 13 platonic solids) to support the object under various test angles.  Archimedes was known to invent devices to verify his mathematical proofs, and most of his experimental techniques remain a mystery.

To reduce the error in discerning fluid levels, Archimedes would have attempted to maximize the fill-factor of the device under test within its sphere of rotation.  It is no doubt, quite interesting, that Archimedes requested that his discovery of the fill-factor of a sphere, within a cylinder, of 2/3rd, be engraved on his tomb.  The dodecahedron structure consumes approximately 2/3rd of the volume within a sphere to which it can be inscribed.

Archimedean solids are highly symmetric, semi-regular polyhedron composed of two or more types of regular polygons meeting in consistent vertices.  A platonic solid only consists of a single type of face.  Archimedes did write a body of work that described them.  However, that work, like many of his contributions, was lost, and not rediscovered until the 1600's by Johannes Kepler.  Of particular interest is the 9th Archimedean solid, the truncated dodecahedron.  It consists of 12 faces, with one decagon, per face.  The 20 equilateral triangles are shared between faces.  Referring to the historical artifact the author has referred to as "sling ammo variant), it can be seen below that the base platform of the Roman dodecahedron is the 9th Archimedean solid:

 

Utilizing the 9th Archimedean solid as a base does not prove Archimedes himself designed the Roman dodecahedron, but it makes him, and those most familiar with his geometric constructs, prime suspects.  Archimedes was well known for his use of circles in engineering design.  In fact, it was reported that his final words were "please do not disturb my circles."  Found on his person, were various mathematical instruments, dials and spheres.  Under the proposed theory, his final words may be interpreted as an attempt to protect his inventions from falling into the hand of his enemy--after all he had been designing contraptions non-stop to deal with his invaders.

If the purpose of the Roman dodecahedron artifact was to characterize sling ammo shape, the design would also fit with the profile of Archimedes life's work.  Some of his greatest war machines, though less discussed by historians, were in the area of artillery such as catapults and stone-throwers.  Historians have discussed how Archimedes created a layered defense surrounding the city.  It began with an outer perimeter of huge stone-throwers that could hurl 500 pound boulders at approaching ships.  With each inner layer, smaller and shorter range weapons were used, such that the Romans were under an onslaught of projectiles across their entire advance.  Unrelenting bombardment is known to have a devastating psychological effect on troops.  The final layered defense at close range was "The Scorpion", a small dart firing catapult that hid behind holes in the wall.  The defenses held until Marcellus ordered a siege and began starving out the defenders of Syracuse.  Even then, the city held out, until finally a traitor of the Greeks let the Romans into the city.

Archimedes wrote the Treatise on Conoids and Spheroids, which finds the volume of solids formed by the revolution of a conic section (circle, ellipse, parabola, or hyperbola) about its axis.  It is the author's belief that the Roman dodecahedron design, utilizing incremental fluid immersion, is the most practical way to measure the volume and sub-volumes of such shapes.  A sub-volume measurement, as has been previously depicted, is repeated below:

It should be noted that not only can an object be rotated at various angles within a bowl--objects may be rotated along its support axis with in the dodecahedron structure.  A perfectly manufactured object, which was symmetric about its revolution of the axis would have an identical displacement profile if rotated along its spindle within the dodecahedron.  In addition, such ammo design may yield a more effective bullet at maintaining aerodynamic spin during long range flight, similar to the American football.  Some sling ammo, however, resembles an almond (perhaps to produce a more consistent lay within the sling), and rotation of he DUT long its axis would have a unique effect on the resulting displacement signatures.

Historians and mathematicians have pondered the experimental methods that may have been used by Archimedes to verify his mathematical treatments.  One known invention is called the Trammel of Archimedes.  It is a mechanism that traces out an ellipse.  It consists of two sliders and a rod which is attached to both sliders by pivots at fixed positions on the rod.  As the sliders move from one side to the other, the end of the rod moves in an elliptical path.  The semi-axes, commonly denoted a and b of an ellipse, are equal to the distances between the end of the rod and the two pivot points (below):

 

 

As shown above, the trammel is a device that can experimentally trace an ellipse.  However, it can also be seen that the angle of the rod with respect to gravity can be varied arbitrarily from 0 to 90 degrees.  Although impractical, it is instructive to consider the situation where one might place the Trammel of Archimedes in a measurement bowl.  It would not be an effective device for varying the angle of immersion to perform displacement measurements of an ellipsoidal body for several reasons.  First, it would require a very large measurement bowl relative to the device under test (low fluid level rise and low fill factor).  Second, the device only works in two dimensions.  The dodecahedron is thus a more practical device for immersion at discrete, fixed, angular intervals.  But what is interesting to consider is that the inventor of the Roman dodecahedron, if utilized for proposed purpose, would likely have conceived of an experimental device that could allow all 3 dimensional angles to be varied within arbitrary increment.  And if the inventor had conceived of such an apparatus, would the timeline have agreed with the proposed invention of the Roman dodecahedron?

The 3-dimensional apparatus that would extend the idea behind the Roman dodecahedron (a cage that can allow 6 unique angles of measurement) to a general angle case, is the Gimbal: 

 

 

If Archimedes was the inventor of the Roman dodecahedron, with utility as proposed, he would likely have considered the generalized case, as his "squaring the circle" method broke 2 dimensional (and 3 dimensional) shapes up into more and more angles, as previously discussed.  He would not have limited himself to the idea of an immersion cage that only allowed 6 angles.  Yes, for the purpose of sling ammo measurement, 6 angles works great.  But he would have pondered a way to rotate an object among all angles, as this is how generalists work.  Therefore, the hypothesis presented lays the case that if Archimedes invented the Roman dodecahedron artifact in his lifetime, he would have necessarily conceived of the Gimbal, and the Gimbal must have been invented during Archimedes time.  It was unknown to the author before, but this testable prediction of the hypothesis, resulted in success.  The Gimbal was first described by the Greek inventor Philo of Byzantium (280-220 BC).  Philo described an eight-sided ink pot with an opening on each side, which can be turned so that while any face is on top, a pen can be dipped and inked - yet the ink never runs out through the holes of the other sides.  This was done by the suspension of the inkwell at the center, which was mounted on a series of concentric metal rings which remained stationary no matter which way the pot is turned.  Philo was born 7 years before Archimedes, and lived 8 years after Archimedes passing.

It is also interesting and relevant to consider the hyperbolic shape of the icosahedron variant.  The asteroid is the common envelope of Archimedes trammel which is moving, and a family of traced ellipses, as illustrated:

 

 SUMMARY OF PROPOSED THEORY