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Geometric Proof of the Red Emerald

For many centuries, every student on Earth was expected to read and master The Elements.

The Elements were The Basics of a standard education, serving as humanity's essential guide to understanding Geometry -- the study of forms, shapes and the world around us.

Euclid's Elements, Definitions, Common Notions and Postulates

The Elements, Euclid's Guide to Defining the Universe

In Thirteen Easy-to-Carry Books

Geometry involves evaluation and definition of physical structures. Developing a methodology to describe three-dimensional (3D) figures has been essential to many professions, but the skill continues to dominate in importance for the modernized workforce. 3D-printing and Computer-Aided Design (CAD) are only beginning to take their logical roles as valuable solutions to the problems of a functioning world. In one of history's stellar achievements, Euclid's great work allows anyone with the ability to read access to this natural power of creation.

No investigation into crystals or gemstones can be conducted without consideration taken for Geometry.

The Roman author Pliny the Elder opened his dialogue on precious gemstones by describing them as "Nature's grandeur presented to us within the narrowest of limits, and in no domain of hers is more Wonder evoked in the minds of many." The Wonder Pliny describes likely originates from the fact pure minerals appear in nature with perfect geometric shapes.

Pyramidal Octahedron with Hexagonal Prism

An uncut Diamond has the shape of two stacked pyramids (Left)

An uncut Emerald has the shape of a hexagonal prism (Right)

A Diamond in the rough has the geometric shape of an octahedron, which looks like two pyramids stacked with a shared base. An Emerald in the rough has the geometric shape of a hexagonal prism, which looks like a six-sided tower. How these perfectly precise angles and impossibly smooth surfaces must have astounded ancient mankind…and they continue to amaze today! In Crystals: Growth, Morphology and Perfection, Ichiro Suragawa provided a comprehensive review of the detailed symmetry found in perfect mineral structures.

In his 2016 presentation How Big, Beautiful Crystals Form, John Rakovan succinctly explained the basic appreciation collectors have for mineral specimens: "The beauty of the symmetry and just the plain fact that they come out of the ground with flat surfaces as if they've been polished [is aesthetic / provokes amazement / inspires Wonder]…[and] even the smallest crystals can have faceted shapes to them in the regular morphology we see."

9 ct red beryl prism with pointed termination

Nine carat red beryl prism with sharply angled termination end

If a red beryl habit displays a single well-formed crystal face, the example is preserved and documented no matter how small in an attempt to add to the collective human knowledge regarding one of the rarest minerals on Earth. Ronald Ringsrud once wrote, "The fascination and Wonder that accompanies the discovery of new knowledge…is exactly the proper use of science: to lead us to amazement of Nature's creation and to experience a sense of Wonder. There is no better place to rediscover Wonder than in [the] remarkable phenomenon that occurs deep inside the finest Emeralds."

The geometric structure of a crystal is affected by three primary forces (Rakovan - 2016):




Let's review the effects of these forces on Red Emerald formation:

Beryl prisms rise from a bixbyite nucleation in topaz-rich rhyolite

Many beryls spring from a rhyolite host rock

richly-mineralized in topaz and bixbyite


(Trace Elements of the Red Emerald) Colored gemstone formation must occur in the presence of TWO elements: a Structural Element to serve as a foundation for the molecule, and a second to act as a chromophore or Chromatic Element -- a metallic ion which absorbs or redirects certain wavelengths of light to create gem color. Diamond famously utilizes carbon atoms in its crystal lattice, but Emerald is formed on a molecular foundation of Beryllium. "Beryl is a relatively rare mineral because there is very little beryllium in the upper continental crust" (Groat & Laurs - 2009).

(The Thomas Range Deposit) Topaz Valley rests on public land less than five linear miles from the Brush-Wellman mining complex, one of the largest and only sources of concentrated Beryllium in the world. Although beryl in rhyolite is uncommon to say the least, Red Beryl likely formed in Topaz-bearing rhyolites due to high local availability of this atomically-unusual element around a great crustal vein of almost incomprehensible size.

The beryllium-rich rhyolite in which Red Emeralds formed came to rest beneath the surface as a lava dome, cooling slowly as heat dissipated (or radiated) in waves, creating an ebb and flow to precise temperature at specific points. A high-temperature magmatic solution holds more ions than that same solution at a lower temperature, and ions are removed as mineral precipitates as the molten rock cools. These precipitates can be reabsorbed if the lava warms again.

Beryllium molecules attach to one another using chemical bonds. Beryl molecules bond side-to-side in two-dimensional nucleus groups. Molecules are added on the same geometric plane, and the group spreads to create a flat, layered surface. The combined molecules ultimately materialize into a thin hexagonal blade one may perceive called a Wafer.

Wah-Wah Red Beryl Gem Wafer Specimen

Although incredibly tiny, weighing under half a carat, the perfect outline of a hexagon can still be seen in this tiny Wah-Wah Wafer

Attachment occurs when ionic saturation is high and/or temperature is low, while detachment happens when temperature increases to allow disillusion of additional ions into solution. When the net overall change is zero, the process of mineral formation is at equilibrium, and the conditions required for further crystallization may no longer be satisfied.

The more physical connections a beryl has with surrounding molecules, the greater number of chemical bonds hold them in place. Stronger contacts resist disillusion, making detachment of a molecule less likely. Molecules stack in three dimensions to increase the number of connections in the group. The surfaces of many islands form at different sizes simultaneously, becoming stable as groups coalesce.

Beryl is bonded layer after layer in this way, causing more and more molecules to become Permanently Incorporated -- buried in a mineral lattice, sharing chemical bonds on all sides with other molecules in the crystal structure.

Thomas Range Red Beryl Gem Wafer Specimen

Layers of stacked hexagons can be counted on the surfaces

of this highly-disturbed and modified Wafer from the Thomas Range


The flux of molecular attachment versus detachment stabilizes the rate of crystal growth so that no single hexagonal group usually outpaces another. The formation of glassy faces is typical, but pressure and movement during formation can create disturbed and uneven surfaces, revealing the operations of independent units. Once a hexagonal group begins stacking molecules faster than others along a preferential vector, the mineral habit extends in that direction to become a Prism.

(Red Emerald Disturbance Modifications) Prisms synthesize with the added pressure of pneumatolytic stress, creating altered beryl crystals which are widely different than typical products of pegmatitic growth. Additional pressure causes a mineral to fight harder against the solid and semi-solid components encountered within the low-temperature environment of a magmatic solution, forcing disruptions to precise molecular order. Formation under pressure also inhibits size and increases damage to crystals.

The majority of red beryl crystals weigh less than half a carat, and the average faceted Red Emerald weighs only seven to eight points (0.075 carat); the largest faceted stone 50 years after discovery weighed a mere 4.5 carat (Shigley - 2003). Whether large or small, specimens exhibit highly-complex crystal structures with recurring morphological patterns, and even the tiniest red beryl specimen can hold clues about the geological conditions present during formation.

Red Beryl grows in wafers, tabs and prisms around Bixbyite

Beryl molecules coalesce around bixbyite seed crystals,

forming wafers, tabular specimens and prisms


(Origin of the Red Emerald) Since "high-temperature (>600° C) lithophysal beryls are tabular as opposed to the significantly lower temperature, prismatic beryls" (Foord - 1996), the general consensus for the Wah-Wah Mountain locality has been that Red Beryl "growth occurred at temperatures below magmatic values (~300-650° C)" (Keith, Christiansen & Tingey - 1994).

Red beryl synthesis is the result of Pneumatolysis -- a process of rock alteration involving high-temperature magma releasing gas under pressure while solidifying. This rarely-occurring natural procedure becomes Hydrothermal at lower temperatures where magma has completely cooled into rock, but "the fact that many fluids escaping from magmas are probably supercritical makes the distinction between liquid and gas somewhat superfluous" (Wones - 1989). Hydrothermal processes are responsible for the crystallization of Green Emeralds in a mica schist.

Crossed Red Beryl twin from Utah and Crossed Green Beryl twin from North Carolina in the Smithsonian Collection

The similar crystallographies of red and green beryl have been documented

The geologic conditions present during formation of the red and green beryl varieties were so similar they produced crystals with the same geometric form, identical modifications to regular morphology and a similar expressed essence.

In The Elements, Euclid's First Common Notion proposes "things equal to the same thing are also equal to one another." If the red variety of beryl has a characteristic modified geometric structure, and if that geometric structure also belongs to the green, then the two varieties enjoy mineral equivalence. Expressed mathematically, if A = B and B = C then A = C.

This rare American gem is therefore geometrically demonstrated to be the Red version of what we call Emerald!


Euclid. The Elements, Ptolemaic Egypt - 300 BC

Pliny the Elder. Naturalis Historia, Roman Empire - 87

Wones, David R. Entry: Pneumatolysis, Encyclopedia of Earth Sciences: Petrology - 1989

Keith, Jeffrey, Christiansen, Eric & Tingey, David. Geological and Chemical Conditions of Formation of Red Beryl, Utah Geological Association Publication 23, pp. 155-68 - 1994

Foord, Eugene. Geology, Mineralogy and Paragenesis of the Ruby Violet Red Beryl Deposit, Southern Wah-Wah Mountains, Beaver County, Utah, Kennecott Exploration Company - February 21, 1996

Shigley, James, Thompson, Timothy & Keith, Jeffrey. Red Beryl From Utah: A Review and Update, Gems & Gemology - Winter 2003

Sunagawa, Ichiro. Crystals: Growth, Morphology and Perfection, Cambridge University Press - 2005

Groat, Lee and Laurs, Brendan. Gem Formation, Production, and Exploration, Elements Magazine, pp. 153-8 - June 2009

Ringsrud, Ronald. Emeralds: A Passionate Guide, Green View Press, pp. 139 - 2009

Rakovan, John. How Big, Beautiful Crystals Form, Dallas Mineral Symposium - August 20, 2016

Rozendaal, Seth. Identifying Similarities in the Crystal Structures of Green and Red Emeralds, Gemmology Today, pp. 88 - May 2017

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