Discussing the mystery of the Costa Rica stone balls. Can we call them spheres? Who sculpted them and what was it purpose? The canquerrique enigma.

 Also you can read  "El pueblo que se obsesionó con la geometría esférica" (spanish but you can translate with google)

 

Background

Costa Rica balls are sculptures dated from 500 B.C to 1500 A.C that were found in the south-eastern pacific region; called Diquís delta, near the Térraba river and Fila Grisera mountains. The culture that sculpted the stones balls will be called in this work as "CANQUERRIQUE people" because CAN means stone in boruca indian language and _querrique is a common suffix used by the huetar precolombinan people in many settlements names. This a personal view of the author of this work, to emphasize that we don't know who were the sculptors. The natives that were found by the europeans about 1500 to 1600 a.c. were not the sculptors (as bruncas, cabécar, cotos etc). About 500-1000 b.c., Canquerricans reached their maximum. They dominated the diquís delta and the surroundings. They reached the Talamanca mountains and the Caño island. Then, in a fast or slow decay, they declined until their disappearance. The continuous flood of the Térraba river, buried the remains of their settlements and its memory. After that, new settlers surrounded the area, as borucas, cotos or quepos.


In 2014, an area around the 17400 m², in the Diquís delta was declared as a world patrimony site. Canquerrique people were organized in chiefdom settlements and transformed gabbro stones, that can be found in the nearby mountains and that have an igneous origin, to beautiful spherical balls. The pottery found near the stones is analyzed by archaeologists to date the balls. It is believed that during about a millennium, canquerrique craftsmen sculpted the stones. After that time, this technical knowledge was lost.

Figure 1. The canquerrique settlers were skilled gold, ceramic and stones craftsmen

Bishop Bernado Thiel was interested in the pre-columbinan cultures and visited the canquerrique region¹²

In 1881, Costa Rica's bishop, Bernado Thiel, a very smart and educated german, visited the boruca territory. He was interested to convert borucas to the catholic religion. Also, he was interested in his culture. He learned precolumbinan languages and also he begun an achaeological collection. He visited the diquis area, particularly he was very close to the ball's sites, near Palmar norte and Pilas (situated some km north from Palmar). He extrated two sculptures from the Pilas, an ownl (with an inverted man head in its peak) and a tapir. Both were sculpted by the canquerriques. Thiel wrote that the sculptors were the "anciant indians", not the borucas. Also, it is important to note that Thiel didn't mention nothing about the canquerrique's balls, even he visited the Terraba river, Palmar Norte and the Pilas. Thiel exposed his collection in Madrid Spain in 1892 including both canquerriques sculptures.

First report of the balls is not from the Diquís region, but from Agua Caliente necropolis

In 1886 (15-september to 3-october), about 40 000 costaricans observed 3 "big" balls and some small ones at the National Exhibition from the archaeological collection of Juan Ramón Rojas Troyo. The origin of the balls was a place called Agua Calientes, an indian necropolis near Cartago city. Fareway from the Diquís area and that maybe connected with the Guayabo culture. Troyo wrote a letter to H. Polakowsky⁸, a naturalist german who was interested in Costa Rica archeology and who had information from Costa Rica's bishop Bernardo Thiel. Rojas Troyo confirmed to Polakowsky⁸ that the balls were found in Agua Caliente site. 9 balls with a maximum weight of 25 pounds (about 11 kg and a diameter of 20 cm). As the balls were found near a empty space, it was supposed by Troyo that was used for some kind of game. He also added in the letter that it would be difficult to the players to move them.  These balls are the only ones that were reported from the huetar region. In 1892, the Troyo collection was exhibited in Madrid and the balls were described as "mathematically spherical". The balls were considered near perfect spheres. Troyo's wife donates the collection to the National Museum of Costa Rica after Juan Ramón death in1887.

Were the Agua Caliente and canquerrique people connected? It was the Rojas Troyo opinion that the people who sculpted his collection was disappeared before 1500 a.c., so there were not the huetars. The same thing happened with the canquerrique people. The pre-columbian people who settled in the surrounding sites, had no abilities to sculpt near perfect spheres and don't knew anything about the canquerrique people. The aguas/calientes-mercedes-guayabo axe and the Diquís settlements began their disappearance about 800 a.c.. Why?

Figure 2. One of the huetar balls from the Troyo collection that was donated to the National Museum.
how this ball came from the canquerrique site to aguas calientes or were the huetars who sculted it? Another mystery

1939 Jorge Lines and Conchita Turnbull expedition ⁹

Jorge Lines was the first anthropology teacher at the Costa Rica University (1950). Conchita Turnbull was the wife of the Honduras UFCO manager that was introduced to archaeology by Doris Stone Zemurray. Both studied ruins in the Sula Valley. Lines and Turnbull visited the Caño Island. They described balls, antromorphic sculptures and metates.  Smalls balls were found in graves from 10 cm to 70 cm in diameter at 1 meter deep. Also, they visited the Térraba river and observed balls with an maximum diameter of 2 m. Also, a ball splitted into two equals parts. Also, they pointed out that the continent balls are bigger than the island ones.

UFCO banana company destroy primary forest and the arqueological sites

In 1939, the stones existence was revealed again during some works of land clearing carried out by the United Fruit Company. This is really a sad date for archeologists and to Costa Rica. The stones were found semi-buried and its exodus from the Diquís delta had also begun without any scientific work and documentation. Stones are now scattered throughout Costa Rica and also worldwide. It was very common in Costa Rica to have the stones as a garden ornament, in private house as well in government building (for example at the National Justice building).

Archaelogical pioneers of the canquerrique culture

Doris Zemurray Stone¹⁻⁷ (D.Z.S., Archaeologist and ethnologist from the Radcliffe College  and daughter of United Fruit Company president) and Samuel Kirkland Lothrop² (S.K.L., Harvard archaeologist) were pioneers in the archaeology of the Diquís Delta.

 

 

Figure 3. The pioneers in the archeology of the Diquís Delta.

1948, Puerto Jesús, Puntarenas. Eleanor Bachman (left), Samuel K. Lothtrop (Harvard's archeologist and Bachman's husband) and Doris Zemurray Stone (Radcliffe's archeologist and daughter of the UFCO president, right).

Next, you can see  infographics for the locations of the balls in its semi-buried or buried state from the archaeologists Doris Zemurray Stone¹⁻⁷ (Figure 3 and 4) in 1940,Samuel Kirkland Lothrop² in 1949 (Figures 3, 5 and 31) and Baudez in 1990 (Figure 7). One can suppose that when a ball was found buried or semi-buried, the word "in situ" can be applied. D.Z.S observed that some of the big spheres that she found have river stone as a support to avoid a movement. In my opinion we need a buried ball and a stone support to use the word "in situ".

In the following infographics, you can observe the most important discoveries in the delta:

Figure 4a. Doris Zemurray Stone researches in the Diquís Delta (1939-1940). 

Map of the farms about 1940 to 1950 of the United Fruit Company (UFCO and called "mamita yunai") in the southwestern of Costa Rica. In this case, UFCO district were divided in 15 farms and sectors (200 m × 500 m) where D.Z.S and S.K.L. explored the Canquerrique culture. Right, some of the Doris Zemurray Stones discoveries in 1940 at the Diquís Delta. Left, a color photo of D.Z.S in 1946, unknown site and sphere, probably a big sphere of 2.14 m diameter at Farm 7.

 

Figure 4b. Doris Zemurray Stone photographic documentation (1939-1940). 

First pics are dated when the UFCO began to destroy the primary forest. Last one before the banana etage began. DSZ show a key element to solve the meaning of the stones, showing a big and small spheres.

 

Figure 5a. Samuel Kirkland Lothrop researches in the Diquís area (1948). 

In Farm 4 (sector 23): Canquerrique culture built platforms with bolder stones. Then a wooden structure is built. S.K.L. found some of these platforms in Finca 4. Three coquine balls were found in these area (this material is unusual). Later, in 1956 after one big flood, 3 graves were found with gold jewelry (about 100 jewels). In 1997 a platform ramp of a big mound was also found with 2 balls at both sides by archeologist Adrián Badilla. Also in Farm 4, but in 36 sector, two spheres triads were found. In the sketch drawn here, distances are in scale (original's Lothrop sketch was really misunderstood because is not in a full scale). The north triad seems to be disconnected with the south one. The north triad was the first in situ sphere park that was shown around 1950 by the UFCO. Some others balls alignment and mound were also shown.

Figure 5b. Samuel Kirkland Lothrop photographic documentation (1948).

The Lothrop misleading configuration sketch of two triads

In Figura 6a, you can see a configuration
that Lothrop considered "in situ". 6 spheres forming a configuration of 2 triads. 3 spheres aligned to the north. The original sketch of Lothrop is misleading because the distance between the balls are not in scale. Lothrop drawn dotted lines to show that. The distances between the balls in north triad aren't in scale; as the distance between the north triad with the south triad. As there is no dotted line in the south triad it seems that it is in scale. But, observing the original photo of Lothrop, the distance between the 1.72 m and 1.42 m balls is 2 m. In the sketch, considering the drawn scale 2.76 m. A new sketch was redrawn using the distance of 2 meters in the south triad, distance between the two triads (26.82 m provided by Lothrop in the sketch), distance between the 1.42 a 2 m balls (6.78 m provided by Lothrop) and finally the angles to build the triangles.

Figure 6a. Original Lothrop sketch of six balls and their pics. Sketch is not in scale

Figure 6b. Corrected Lothrop² sketch in scale as must be

In this link, you can see an introduction of the canquerrique scultures and the Lothrop configuration of 2 triads: The Lothrop configuration of 6 spheres in the Diquís Delta (spanish)

In figure 6b, you can observe the real configuration of the six spheres. In my opinion, the south triad seems not to be connected with to the north triad and the balls are really close. There is no evidence that the spheres have a stone support. The north triad was not discovered by Lothrop but by the UFCO and it was exhibited about 1947 as a park ball by the manager. 

 Latest discoveries in the Diquís

In 1990, the french archaeologist Claude Baudez (an outstanding mesoamerican archeologist), found 5 spheres semi-buried in farm 7, today called farm 6 and world wide patrimony site. Later, archaeologists of the Costa Rica National Museum in 1993 (Corrales and Badilla), near the location of these spheres, found a full buried  sphere next to a mound and a ramp.  In 2005, the other sphere at the other side of the ramp was also found (Adrián Badilla). In Figure 7, you can see these discoveries in Farm 6 (not in scale in this case, this is the old farm 7). Adrían Badilla, Francisco Corrales and Ifigenia Quintanilla are the costarican archaeologists that investigated the delta in the last 30 years to crack the mystery. 

Figure 7. Reconstruction of a settlement in Farm 6 (old Farm 7).
The sphere sizes had been augmented to be seen in the sketch.

As you can see in Figure 7, balls were placed in front of the Pre-Columbian houses to show a social status or an important social role of the owner of the house, as to be the Chieftain of the settlement, or an spiritual leader (shaman), but also in public areas.

In the United States, you can find canquerrique sculptures in front of the Peabody Museum in Boston since 1964 (109 cm and 10.54 tons). At the Metropolitan Museum of Art (in New York, 66 cm diameter and 385 kg mass). Two others in Fairmount Park Association (Philadelphia, 4000 kg and 10 700 kg (1.85 diameter) and showing some cracks). At the garden of the Costa Rica embassy in Washington D.C..

Some general conclusions and remarks:

1- Time, huaqueros and the UFCO destroyed most important evidence to crack the mystery.
2- Supposed spheres "in situ" because they were found buried, can not really be in its original positions. Only spheres buried and with a stone support can be "in situ"¹º.
3- Configurations are a noise added in the search to crack the code as the Lothrop's sketch.
4- The primary purpose of the sphere is spiritual or religious. As the sizes changes from some centimeters to meters, what is important is the spherical shape, not it size. This geometrical shape is considered as perfect in the sense that have infinite planes of symmetry. For a fixed volume, the spherical geometry have the lower external area.
5-Jorge Lines described graves with spheres in the Caño Island in 1939⁹. Lothrop wrote a testimony where it was found balls in 2 graves in Farm 4 and sector 23 (see Figure 5,). The most important gold treasure that was reported was found in this site. Badilla also found an small ball in what is supposed to be a grave.
6-Smaller balls (below 1.5 m) were found on top of the mound and in its surroundings (particularly at both side of the ramps with a 1 m size). The bigger balls were not found in front of the chiefdom or shaman dwellings, but in public areas (above 1.5 m) as you can see in Figure 7. Collectivity is more important than individuality.
7-Secondary purpose is to move the spheres as a territory mark. Canquerrique sculpted big spheres (above 1.5 m) to mark his territory and to show his power when they got involved in wars to defend the rich agricultural lands in the delta.

How close are the balls to an spherical shape?

1-Introduction

Figure 8: Applying two mason levels to estimate the diameter

Archaeologists usually applied a simple technique to measure the diameter of the balls as you can see in Figure 8.  Two mason levels play the role of two mathematical tangents. The sphere diameter is the distance between the two tangents. This method was applied first time by Lothrop² in 1948.


In 2013, Lanamme⁴⁻⁵, a Laboratory of materials located at the Costa Rica University, applied a 3D scanner to measure the diameter of two stones and to obtain a 3D model. One ball was located in the garden of the Costa Rica University in San Pedro (in from of the Agri-food Science Faculty). It was purchased in 1970 to the United Fruit Company. The other one was located in the Silencio at the Diquís area, near Palmar Norte. It is the biggest stone found until today,

They get a precision of 3 decimals in their measurements. They found that the San Pedro stone has a diameter of 1.965 m with a maximum variation of -0.005 mm with regard a perfect circle, that is to say an error of 0.25% (in diameter). An outstanding precision for the canquerrique sculptors. The El silencio stone was estimated with a 2.66 m diameter and 3D model was developed to study its changes with time due to its high deterioration.

Figure 9: Lanmme measurement estimation of the diameter appling a 3D scanner

The San Pedro stone is a typical one, a coarse grained polished surface, gray color with some dark regions (plagioclases and hornblendes) and sculpted in a volcanic material (gabbro in this case, but also sometimes spheres were crafted in limestone or sandstone). A very beautiful sculpture due to its symmetrical shape to the naked eye.

In the next paragraphs, a mathematical and technical method were proposed to calculate sphericity factor, and to answer how symmetrical are the sculptures.

 

2-Measuring sphericity (ψ)

Sphericity is defined by Wadell⁶ as:

Where Vp is the volume and Ap is the area of the object. For example, the sphericity factor for an sphere is 1 and for a cube 0.806. Sphericity is a comparison ratio between the area of an ideal sphere calculated with the volume of the object (numerator) and the real area (denominator). In order to applied this equation, you need the surface and volume measurements.

Consider now an spheroid with three semi axes a, b and c. When a=b=c, an spherical shape with a 1 factor is obtained. When a=b  and c≠a, an oblate spheroid is obtained; the same shape of the earth. The area and volume can be calculated with the following equations for a 3D ellipsoid having 3 different semi-axis a, b and c, as follows:

 

where V is the volume and S the surface. p is equal to 1.6075. Replacing equations 2 and 3 in equation 1, results the following equation:

same value for p (1.6075). The ψ spherity in equation 4 is one for a perfect sphere. In equation 4, you need the three semi-axis measurements to calculate sphericity.

Consider in equation 4, an oblate spheroid, where a≠b and c=b. Then, equation 4 simplifies to:

3-Measuring circularity (ɸ)

When a plane cuts an sphere, the resulting intersection is a circle area. So, it is useful to define circularity as follows:

Where A is the area and P the perimeter of an ellipse (with semi axis a and semi axis b). For a perfect circle ɸ=1. Circularity is a comparison ratio between the real area of the object (numerator) with an ideal area of a circle calculated with the perimeter of the object (denominator). In equation 6, you need the area and perimeter measurements of a plane cut in a sphere. You can get them with a photo pic of the sphere and image processing software.

Using the Ramujan approximation for the ellipse's perimeter in equation 6, one can obtain:

 

In equation 7, you need the area and the semi-axis measurements of a plane cut in a sphere. You can get them with a photo pic of the sphere and image processing software. For example with Photoshop you can calculate area, vertical and horizontal distance.

4-Mesuring the aspect ratio (roundness)

The roundness is defined as follows for a 2D ellipse:

where and and b are the semi-axis of the ellipse with the condition a ≥ b. A circle have the property that ℛ=1.

5-Comparing the sensibility of sphericity, circularity and roundness with regard the shape of a circle.

In figure 10, it was calculated the 3 parameters (sphericity, circularity and roundness) for different cut's plane of an spheroid. The yellow figure is the circle (a plane's cut of an sphere). Orange and red figures are the cut's plane of an oblate spheroid. It was noted, that the roundness factor is the most sensitive to changes in the size between the three parameters. For a reduction in the b semi axis of 21%, the roundness factor changes 28% (Red ellipsis, the semi axis (a) was not changed). For the same reduction in the semi axis b, a change of only 1.2% was observed in the sphericity parameter and a 2.5% change in circularity one is obtained. These two shape parameters are rigid with regard changes in the rounded shape.

Figure 10. Impact of shape in sphericity, circularity and roundness parameters.

Yellow circle is the projection of an sphere. Orange and red are projections of  an oblate spheroid.

6-Applying the Lanamme results to calculate sphericity (equation 5) for the San Pedro ball.

Lanammes⁴ measures a diameter of 1.956 m and an error of  -0.005 m with regard to a perfect circle. So, the diameter is between 1.956 and 1.951 m for the San Pedro sphere. Sphericity can be calculated applying equation 5 with:  a=1.956/2, b=1.951/2 and c=a=1.956/2, obtaining a factor of 1.00 (near perfect). Applying equations 2 and 3, it was obtained the following values: V=3.9084 m³ and S=11.9991 m². Considering for gabbro a 3000 kg/m³ density, then the sculpture is about 11 725 kg (about 12 tons).

Also Lanamme⁴ published the values of area (12.28 m²) and volume (3.918 m³) and a mass from 10.5  to 12.95 tons (due that the density is not really known). Replacing this data in equation 1, an sphericity value of 0.98 is obtained (Lanamme's didn't report the sphericity result). The drawback for the Lanamme's study is that a portion of the sphere was hidden in the support area. Some results have been rejected due to this unknown region. The 3D models for the sphere obtained in this work has to be similar to sphere shown in Figure 14, with an unknown area in the base. How they obtained area and volume?

 
 7-Some stadistics about the spheres

In Figure 11, you can see the graphic of the frequency of the spheres versus its diameter in the Diquís delta, for the Doris Zemurray Stone¹ expedition in 1940, for S. K. Lothrop² in 1950 and other known spheres. A total of 74 spheres. Most frequent size is about 0.67 m. There are few spheres with more than 2 meters. The maximum size is the green sphere that correspond to the Silencio with 2.66 m. This sample is about 25% of a total of 300 spheres that are supposed to exist. The data size of Stones were less accurate than those of Lothrop. 

Figure 11. Frequency vs. diameter of the sphere.
  Data of 74 spheres from: D.Z. Stone (1939, 24 yellow spheres), S.K. Lothrop (1948, 40 red spheres) and other spheres. Green sphere is the Silencio. It is supposed a total of 300 ball.

8-Calculating sphericity, circularity and roundness with the Lothrop² data

Lothrop² measured the diameter of the spheres using a tape and a plumb bob as you can see in Fig. 8. He also measured the circumferences that corresponded to different plane cuts of the spheres. He measured the circumferences for the sphere equator and for 4 obliques circumferences. In Fig. 12, you can see the equator circumference and two obliques circumferences that pointed out to the east and to the west.  These two oblique circumferences are separated by 90°. He measured two others obliques circumferences pointed out to the north and south. In order to calculate the sphericity parameter, the semi axis a, b and c were obtained calculating the radius from three circumferences of the Lothrop² data. North circumference for the semi-axis a, equator circumference for semi axis b, and south circumference for the c semi axis. You can see the values of a, b and c in Table 1. Sphericities were calculated with these data applying Eq. 5. Circularities were calculated applying Eq. 5 with the semi axis a and b. In Eq. 5 the area was calculated with the formula for an ellipsis: A=abπ. Roundness was calculated applying the Eq. 8. All these results are tabulated in Table 1.

Figure 12. The Lothrop method to measure sphericity.
Lothrop measured the circumference for three plane cuts of the sphere pointing to the north-south axe (three orthogonal semi-axis).

Table 1. Results of sphericity, circularity and roundness for the Lothrop data.

In the first column of Table 1, it were tabulated the location of the spheres: first the farm (F), next the sector (S) and finally the sphere (A, B, C, etc.). These ball stones, as you can see in the results of sphericity, circularity and roundness are close to 1 (value for an sphere).

9- Two proposed methods in order to measure sphericity of the Diquís sculptures.

Two graphical methods are proposed is this section to measure sphericity and circularity: the one and two shots methods.

 A-Two shot method to estimate sphericity

1-Take two shots of the sphere where you can see all the contour as it is showed in Figure 13. The shots are 90º separated, same distance from the sphere and situated in its center. You can take the shot at two different distances if you have an object of known size (a reference one). In this way you can get the absolute value of diameter. Or to take the shot to the same distance from the stone having only relative measures (with pixels units in Photoshop).

Figure 13: Method of two shots to measures sphericity.
Estimate parameters a, b, c for an ellipsoidal shape (orthogonal semi-axis).

2-Process the photo to obtain the measures a, b and c.

Using the Photoshop software, it is possible to obtain from a pic the contour of the sphere. Using the analysis option in Photoshop one can obtain the area of the circle and the major a minor axis of the circle from the two shots (horizontal and vertical axis). These shots were selected in these angles because the low part of the sphere is hidden.

3-Finally, apply equations 1, 2 and 3 with the measurements a, b and c in order to obtain the sphericity (eq. 4), circularity (eq.7) and roundness (eq.8).

B- One shot method to estimate sphericity and circularity.

In the case where only one shot is available, an approximation of the sphericity can be obtained with a≠b and c=b. In this way sphericity can be obtained with equation 5, supposing an oblate spheroid. Also, you can calculate circularity applying equation 7. The contour of an sphere obtained in one shot is equivalent to a plane cut of an sphere that results in a circle.

 

Figure 14. Graphic one shot method to estimate sphericity.

Sphere and camera configuration in order to get a full circular cut. Note that the low part is hidden as you can find for the majority of the spheres, in the contact area with the soil. Few spheres are in pedestals.

 

1- Take a shot with an angle where you can see the full circle contour of the sphere. Note that, sphere have a hidden section in the part that have contact with soil. See Figure 14.
 

2-Process the photo in Photoshop and measures area and the semi-axis a and b.

3-Calculates sphericity, circularity and roundness with equations 5, 7 and 8.

4-You can take more shots and repeat the process to get a more accurate result.

C-Applying the one shot graphic method

1- Testing the method with the UCR ball (Agri-food faculty, since 1970)

Four pics were taken to the ball located at the Costa Rica University (UCR), in front of the Agri-food Faculty. The same that the Lanmme³⁻⁴ test with a laser scanner. First picture was taken pointing out to the north, second picture to the S45°E, third one to the E and the fourth one to the N45°E. The angle of the camera was about 30° from the equator plane to avoid the support section. In Fig. 15 to 18 you can see the resulting pics and also the results of area, vertical axis and horizontal axis. These results were tabulated in Table 1. Applying equations 5, 7 and 8 with these data you can compute the values of sphericity, circularity and roundness that were also tabulated in Table 1.

Figure 15. View S45°E

Figure 16. North view

Figure 17. East view.

Figure 18. N45°E

Table 1. Results of sphericity, circularity and roundness for the ball of Agri-food Faculty in UCR.

Remember that the parameters of ψ, ɸ and ℛ are dimensionless. The result agrees with the Lanamme³⁻⁴ results as they are close to 1. Most sensitive parameter to the shape is roundness. The maximum error in roundness for the ball is 2% in the east view.

2-Testing the method with the stone ball at the Costa Rica Museum in the central yard.

The ball, located in the central yard of Costa Rica Museum, was selected in the second test for the graphic method. This ball is about 1.8 m, have observable errors in symmetry and different level in the polishing quality. It is possible that this ball was unfinished. The less symmetry side of the ball is shown in Figure 19. This picture was rotated to take 3 measurements of the area and axis. These results are tabulated in Table 2. Finally, the sphericity, circularity and roundness were calculated and presented in Table 2.

Figure 19. Ball at the Costa Rica Museum. Central yard.

Table 2. Results of sphericity, circularity and roundness for the ball of the Costa Rica Museum, central yard.

You can see that roundness is the most sensitive parameter. In this case about a 6% of error was obtained. Some changes are observed in sphericity in the fourth decimal, a some in the second decimal for sphericity.

3-Museum Balls

Some of the most beautiful (symmetrical) spheres can be find in some museums in Costa Rica and United States. In this study, it was selected six symmetrical spheres (to the naked eye) from different parts of the world to calculated its sphericity, circularity and roundness applying the one shot method.

From Figure 20 to  25,  you can see the pictures of the studied spheres with the processing image of Photoshop.

Figure 20. Sculpture 1. Sphere of the Metropolitam Museum of Art. Purchased in Costa Rica circa 1960.
 Notice the Photoshop analysis result: area, horizontal and vertical axis measurement (all in pixels)

Figure 21. Sculpture 2. National Costa Rica Museum. About a diameter of 1.9 m

Figure 22. Sculpture 3. Auctioned sphere by Artemis gallery' purchased in 1960 in Costa Rica.

Figure 23. Sculpture 4. One of the two sculptures in Fairmount Park from Palmar Sur. 1.8 m and 9 tones.

Figure 24. Sculpture 5. Sphere located at the Peabody Museum (Boston). 
About 1.01 m and 2 tones. Origin Farm 2, South Palmar Sur. Donated by CR gov. (Doris. Z. Stone)

 

Figure 25. Sculpture 6. Costa Rica National  Museum collection.
 This photo was shotted at th exposition int the Quai Museum (Musée du Quai in Paris).

In the next table you can see the results of the sphericity, circularity and roundness applying the graphic method of one shot.  Area and diameter weres obtained processing the pics. Next, sphericity, circularity and roundness shape parameters were calculated with equation 6, 7 and 8 respectively.

Table 3. Results for six museum spheres of sphericity, circularity and roundness. For sphericity and circularity, the results were truncated to 1 decimal, but in all the cases the numbers have three nines.

4-Conclusions

These results confirms that some of the spheres, that seems to be very symmetrical to the naked eye and that were selected by the museums as beautiful examples of the Costa Rica sculptures, are nearly perfect spheres.

The Peabody museum (Boston) have the opinion that is better to call the stones as balls, quoting the museum:

Some people refer to these objects as "spheres," but since all are not perfectly round and "sphere" generally refers to a hollow form, I believe that "ball" is a better term as it does not pretend to a precision that is not always present.

The measured shericity, circularity and roundness shape parameters shows that the adjective spherical for these stones can be apply as a recognition to the skilled canquerrique craftsmen . There are many types of stones, some are made in limestone but majority are gabbroid. The size varies from some centimeters until the 2.66 meters of the Silencio Stone. About 300 stones was estimated to exist. Many of them are not so perfect as the San Pedro. But it is amazing how some of the Canquerrique sculptors have reached a such perfection in his art with rudimentary stones tools.

Last words

In Figures 26 to 29, one can observe that the precolumbinan people was skilled craftmen. Anthropomorphic and zoomorphic sculptures can be observed in Figura 26 with the huetar style. A ball with the canquerrique style also can be observed (not an sphere). All sculpted in granite. Figures 27 to 29 are artistic work from the diquís area. In Figure 27 a cylinder call barrilete is shown. In Figure 28 an excellent sample of the jewerly in the region can be seen. Figure 28 shows a head trophy sculpture. 

Figure 26. A set of 4 stones sculptures with a huetar and canquerrique (ball) styles

 

Figure 27. The barriletes are stones with a cylindrical shape.
Three views in photo of the same barrilete (a height of 50 cm). From the collection of the Costa Rica National Museum.

Figure 28. A Gold ornament typical of the Diquís.
Gold collection from Costa Rica National Museum.

Figure 29. A canquerrique warrior with a head trophy. 
The sculptor designed a pedestal to support the statue in from of the dwelling. This feature is unique of  the canquerrique culture.

During about a millennium, an unknown culture was settled at the diquís delta, called in this work Canquerrique. They prioritized collectivity to individuality. Their spirituality believes and world vision were reflected and enhanced in the spherical shape. Around 300 b.c., skilled artist developed a technical to sculpt and move huge blocks of granite until a maximum of 2.66 m. Small balls (lower than diameter 1.5 m) was placed on top of the mounds, in its surroundings, both side of the dwellings ramps and graves. Big balls (below 1.5 m) were placed in public areas. With time, spherical balls were an icon of his power and identity. Balls were used to mark their territory. This society reach a maximum about 800 a.c, and then they declined until no descendants were found about 1500 a.c. 

 
Why canquerrique people declined? how they sculpted the balls? Where was the granite source? Is there a connection with huetar people? Where canquerrique people came from? Why some balls are twins if they weren't place next to a ramp (as Lothrop's triads)? Hope, archaeologist in the future find more keys to crack the mystery of the canquerrique people.

Bibliography

1-D. Stone . (1943). A Preliminary Investigation of the Flood Plain of the Rio Grande de Térraba, Costa Rica. American Antiquity, 9 (1), 74-88.

2-S. K. Lothrop. (1963). Archaeology of the Diquís Delta. Papers of the Peabody Museum of Archaeology and Ethnology, Harvard University. Vol. LI. Cambridge, Mass.

3-E. Lothrop. (Sep. 1955). Mystery of the Prehistoric Stone Balls, Natural History.

4-Paulo Ruiz, José F. Garro, Gerardo J. Soto. (Jul./Dec. 2014). The use of Lidar Images in Costa Rica: Case Studies Applied in Geology, Engineering, and Archeology.  San Pedro de Montes de Oca.  N° 5, Rev. Geol. Amér. Central.

5-F. Garro, P. Ruiz and K. Herrera. (2013). Lanamme-UCR realiza escaneo 3D de esfera precolombina. San Pedro, Montes de Oca, Lanamme report.

6-Hakon Wadell. (Jul. - Aug., 1932). Volume, Shape, and Roundness of Rock Particles.  The Journal of Geology, 40, n° 5, 443-451.

7-Stone, D. Z. (1961). The Stone Sculpture of Costa Rica. Cambridge, N° 13, Essays in Pre-Columbian  Art and Archaelogy, Massachusetts, Harvard, University Press, S.K. Lothrop.

8-H. Polakowsky. (1892). Antigüedades de Costa Rica. Anales del Instituto Físico-Geográfico y del Museo Nacional. 

9-Jorge Lines y Conchita Turnbull. (1938)  Informe de la expedición arqueológica Turnbull-Lines en la región brunka. Enero-Febrero 1938.

10-Doris Stone Zemurray. (1954). Apuntes sobres las esferas grandes de piedra halladas en el río Diquís o Río Grande de Térraba en Costa Rica. Informe interno al Museo Nacional de Costa Rica. 1954 (Boletíno informativo MNCR, Año 1, N° 6).

11-Mario Llosent Rescia. (2019). El misterio de los Canquerrique. Escultores precolombinos de bolas de piedra en Costa Rica (s.f.). Recuperado de https://fikismatica.blogspot.com/2019/07/parte-1-las-esferas-de-lothrop-en-la.html

12-Elías Zeledón Cartín (2003). Crónicas de los viajes a Guatuso y Talamanca del obispo Augusto Bernardo Thiel. 1881-1895. Editorial de la Universidad de Costa Rica.