Larger weights. Larger weights have more sides.
20-dirham weight with 42 sides.
Two octagons and 40 smaller sides in eight diagonal rows of five each (third photo).
58.2 grams. 23-20 mm. (2.91 grams/dirham)
There are also rarer 50-dirham polyhedral weights.
The great majority of the 5-dirham polyhedral weights do not have any Arabic legend. Next is the most common design: two hexagons with three rows of sides between them and all sides decorated with bird's eyes.
5-dirham weight with two hexagons and 18 smaller sides in six diagonal rows of three.
14.77 grams. 12 mm between the hexagons. 17 mm maximum. (2.954 grams/dirham)
"Bird's eyes" on every side. On this example the central depressions on the hexagons are deep.
Other Shapes. Small weights are often rectangular, which are usually not distinguishable by shape as Islamic as opposed to Byzantine or Roman. However, there is another shape, the barrel, which is likely to be Islamic if the mass is right, even though weights of that shape have also been found in Roman contexts, including at Pompeii.
Barrel weight. 5-dirham.
14.90 grams (2.98 grams/dirham). 15.2-12.2 mm.
The ends are very nearly circular, making the cross-section also nearly circular.
Different shapes. Some Islamic weights are polygonal, but with different shapes than the more-common examples above.
5-dirham weight with 14 sides--six square and eight triangular--
in the shape of a cube with all eight corners lopped off so that each forms an equilateral triangle.
14.66 grams (2.932 grams/dirham). 16 mm maximum and 12.6 mm minimum.
On this example each side has a shallow incuse rectangle with something in it Maybe they have letters as designs. They are all similar to each other but do not all seem to be the same and none are clear to me.
For short, this shape could be called a "truncated cube" (although it is a different shape than the 2-dirham truncated cube above). It is possible it is not from the same series as the polyhedrons above and it may be from a later Islamic or Ottoman period. However, the mass-standard is apparently the same.
2-dirham weight with 12 sides.
Four squares and eight triangles.
5.90 grams (2.95 grams/dirham). 12.0-9.6 mm.
This shape could also be called a truncated cube.
Islamic polygonal coin weights are common. Holland (1986) published a list of over 600 weights which were surface finds from the neighborhood of Caesarea Maritima (in Israel, on the coast north of Tel Aviv), most of which were lighter brick-shaped weights and 83 were polyhedral. That's just surface finds from one medium-sized city! Apparently, every medieval city in the region, which included Egypt, has weights in abundance. Until the last few decades locals just threw them away as having no value. Now some auction firms are selling them, but prices are very low. Holland wrote, in WWOCM, Chapter 8:
"Of all the different sorts of weights to be found at Caesarea Maritima, the Islamic bronze are by far the most numerous. I have examined more than two thousand, the great majority of them unmarked. The inscriptions on some of the weights (many more await decipherment) link them to the Fatimid rulers of Egypt. This would date them (roughly) to the XI – XIII centuries C.E.
"All of the weights fall (according to mass) into one or other of two series. One series is based on a unit of about 2.9 g. The unit of the other series is about 4.2 g. The two series of weights are therefore often referred to as the "dirham series" and "dinar series" respectively, even though during the period mentioned above (XI –XIII cent. C.E.), these mass-standards were no longer strictly applied to Islamic coinage. It is generally supposed that these weights were used for weighing precious metals --- bullion and/or coin. This supposition is based entirely on circumstantial evidence."
"Flinders Petrie, whose weights were acquired largely in Egypt, does not mention these small weights at all (His own remark, regarding weights in general, is that the fellahin who supplied him considered them worthless, and threw them away, until they were offered a reward --- after which he acquired more than five thousand specimens.)" [Note: This is about all weights, not just polygonal weights. Petrie's comments are from long ago--near the 1920s.]
What were weights for? When the silver dirham denomination was first issued c. 90 AH (c. 708 AD) it was good silver and weighed c. 2.93 grams. This standard was maintained for coins for many years. However, by the time the Fatimid dynasty took Egypt and the Levant in the early 10th century, silver coins were no longer produced with a consistent mass. Fatimid silver coins are mostly half-dirhams which are rare (and expensive) compared to earlier Islamic silver coins. According to Holland, every transaction involving silver must have had the silver weighed, apparently using the old dirham standard. Holland (WWOCM) notes the mystery that while the weights are very numerous, balances and even parts of balances are extremely rare.
I wonder if these weights were used to weigh market items such as spices, wheat, or other produce?
More examples. Some weights are similar to the polyhedra above but without the "bird's eye" pattern in the polygons and the with the sides more rounded. Holland calls this design a "peeled orange" [WWOCM, Chapter 8, page 150]
10-dirham weight with two octagons and 24 smaller nearly-square sides in eight vertical rows of three sides each. Each side is rounded making this weight almost a truncated sphere.
"Bird's eye" pattern only in the octagons.
29.22 grams (2.92 grams/dirham). 13 mm between the octagons. 19 mm maximum.
Holland notes that all the "peeled orange" examples he has seen have octogons.
5-dirham weight of "peeled orange" shape with two octogons and 24 smaller rounded sides in eight vertical rows of three nearly-square sides each. Each side is rounded making this weight almost a truncated sphere.
"Bird's eye" pattern only in the octogons.
14.40 grams (2.88 grams/dirham). 12 mm between the octogons. 15 mm maximum.
Some polyhedral weights have "bird's eyes," but fewer.
5-dirham weight with two hexagons and 18 smaller sides in six diagonal rows of three sides: pentagon-square-pentagon.
"Bird's eye" pattern in the squares but not the pentagons.
14.49 grams (2.90 grams/dirham). 12 mm between the hexagons. 15 mm maximum.
Most 5-dirham weights have two hexagons and six rows of three smaller sides, but some are different:
5-dirham weight with two decagons (10 sides) and 20 smaller quadrilateral sides in ten vertical rows.
14.96 grams (2.99 grams/dirham). 12 mm between the decagons. 15 mm maximum.
I don't know (and I don't think anyone else does either) if these different shapes are from a different time-period or region, or have any special meaning at all.
10-dirham weight with two octagons and 16 smaller sides in eight vertical rows, each with two four-sided sides.
Only one (very weak) "bird's eye" pattern in one of the octagons.
29.82 grams (2.98 grams/dirham). 16.8 mm between the octagons. 20 mm maximum.
This one is similar, but has a legend on one of the octogons. The legend seems to be "la ilah illa"
"There is no God except Allah".
10-dirham weight with two rounded octagons and 16 smaller sides in eight vertical rows, each with two four-sided sides.
29.59 grams (2.96 grams/dirham). 16.6 mm between the octagons. 20.6 mm maximum.
Other weights. Most Islamic weights are denominated in dirhams (the dirham was a silver coin denomination), but a small fraction of weights are denominated in dinars (the gold coin denomination). The gold dinar originally weighed c. 4.55 grams, close to the weight of a Byzantine solius, but the reform of Abd al-Malik in 693 changed it to c. 4.25-4.19 grams where is stayed for centuries. However, unlike Byzantine solidi, Arabic gold coins have widely varying weights and must have been weighed individually for each transaction. Weights on the dinar standard are squat and "discoid." Dinar barrel weights are like thick disks, proportionally wider and shorter than the dirham barrel weights.
An Islamic 5-dinar (not dirham) weight.
19 mm in diameter and 9 mm high. 20.81 grams.
(4.162 grams per dinar).
This dinar-standard weight is "discoid," unlike dirham weights.
Another discoid weight on the dinar standard. This one is 1-dinar.
13.1 mm diameter and 4.4 mm thick. 4.06 grams.
Next is an polyhedral weight which is unusual for being so small. Weights with light masses are very common, but usually with brick shapes and not polyhedral shapes.
Weight with 18 sides--two octogons and two horizontal rows of eight pentagons.
8.3-5.3 mm. 2.05 grams. The sides are much smaller than on the 5-dirham examples.
It is much too light to be a 1-dirham weight (c. 2.94 grams) and it has the squat disc-shape of the dinar series. Its mass is very close to a half-dinar. (Half of 4.2 grams is 2.1 grams.) I think it is a half-dinar weight.
Which standard? Weights that do not conform to these standards might simply be poorly or criminally made, or worn and corroded, but they might simply be from a different culture or region, especially if they lack the bird's eyes.
A barrel weight.
18.6 mm wide. 16.2 mm tall. 27.88 grams.
27.88 grams is not far from a 10-dirham weight which should be 29.4 grams, but the close conformance of polyhedral weights to the dinar standard suggests this weight is not from the same series as the polyhedral weights above.
Maybe it is Roman or Byzantine. Could it be a one-ounce weight using the Roman pound? The weight of the Roman pound is subject to scholarly dispute, but is probably near 327 grams in the early empire and lighter later. The corresponding ounce would be 27.25 grams. So, maybe this weight at 27.88 grams is slightly overweight or maybe it isn't a Roman one-ounce weight.
I am accepting suggestions, information, and supporting evidence. Write me at 
.
Uncertain denomination. Uncertain culture. Uncertain era.
This shape is hard to describe. Maybe "Two truncated cones back to back."
30.22 grams.
21 mm wide and 13.4 mm between the flat sides.
It is close to round making the sides hard to count. The middle image suggests 11 sides and the left image is too rounded to count.
The 10-dirham denomination tends to be close to 29.4 grams plus or minus a few tenths. At 30.22 grams this weight is outside the usual range.
For more examples (but not more discussion) see the second page. A24156
Conclusion. Islamic polyhedral 5- and 10-dirham weights, probably from the Fatimid era (10th-13th C.), are common from Syria, Lebanon, Jordan, Israel, and Egypt. Unfortunately, there are few scholarly sources about them. Their masses are notably regular and correspond to a dirham of c. 2.94 grams. There is no published contemporary documentation of their uses. They may have been used to weigh silver for transactions, but evidence for that is lacking other than the fact that the masses fit.
References: (In order of usefulness--not alphabetically by author.)
Goodwin, Tony. "Medieval Islamic Copper-Alloy Money Weights from Bilām d al-Sham" in INR 7 (2012) pages 167-180. 25 weights pictured throughout and 23 on plate 18. About half are these types of polyhedral weights.
[WWOCM] Holland, Lionel. WWOCM Chapter 8. "Islamic Bronze" 2009. [11 unnumbered pages. The best source I know of] Available on Academia.edu:
https://www.academia.edu/3596498/WWOCM_Ch_8_Islamic_Bronze
Holland, Lionel. "Islamic Bronze Weights from Caesarea Maritima," ANSMN 31 (1986) 171-201 and plates 33-36.
https://www.academia.edu/3542895/Islamic_Bronze_Weights_from_Caesarea_Maritima [Page 182 gives the numbers of each type]
(This artcle has photographs of 13 dirham weights and 4 dinar "discoid" weights shaped like these. In addtion, it has 4 (not truncated) cubes, 7 small discoid dinar weights, 5 small brick-shaped weights, and then 58 small brick-shaped weights with some inscription.)
Holland, Lionel. Weights and Weight-like Objects from Caesarea Maritima (WWOCM) 2009. Chapters 1, 2, 3. Available on Academia.edu:
https://www.academia.edu/3596231/Weights_and_Weight_like_Objects_from_Caesarea_Maritima_WWOCM_Chapters_1_2_3
Balog, Paul. "Islamic Bronze Weights from Egypt," Journal of the Economic and Social History of the Orient, 1970, pp. 233-255, available on Jstor: https://www.jstor.org/stable/3595979 [27 images, mostly of barrel-shaped weights]
Hendin, David. Ancient Scale Weights. Hardcover. 239 pages. These polygonal Islamic weights on pages 225-229. [Almost all of his information cites Holland.]
On-line resources:
"Balance Weights in the Collection of the Anatolian Civilizations Museum in Ankara" by Oğuz Tekin
https://journals.openedition.org/anatoliaantiqua/383?lang=en [Many of the illustrated polyhedral weights were found at Osmanyie at the northeast corner of the Mediterranean Sea.]
A collection of weights (not all polyhedral):
https://www.forumancientcoins.com/gallery/thumbnails.php?album=5252
References of lesser relevance.
Rebsrock, Ulrich. "Weights and Measures in Islam," in Encyclopaedia of the history of science, technology, and medicine in non-Western cultures. 2008, pages 2255-2267. Available at Academia.edu:
https://www.academia.edu/26399908/Weights_and_Measures_in_Islam?email_work_card=title
[No discussion of these polygonal weights. This is about the relationships between named units, not the physical artifacts. Most weights discussed theoretically are for dinars, that is, gold, not silver dirhams. The measures were different in different parts of the Islamic world. Maybe the locations and time-periods of these polygonal weights could be inferred if we knew the silver weight standards in various locations at various times.]
Bendall, Simon. Byzantine Weights: An Introduction. 68 pages. Paperback pamphlet. [This work does not consider Islamic polygonal weights]
For more examples (but not more discussion) see the second page. A
Go to the main Table of Contents page for this entire group of educational pages about ancient coins.
Islamic Polyhedral Weights 
A 10-dirham weight with two octagons and 18 smaller sides in six diagonal rows of three sides each.
A 2-dirham weight with 14 sides. It is a truncated cube. 
A 2-dirham weight. Brick shaped. 
A 1-dirham weight.