An octagram is, to put it mildly, an eight-angled star polygon . A shape that, much like certain individuals, insists on being noticed through its self-intersecting nature. If you’re here, presumably you’ve managed to grasp that much.

Regular Octagram

It’s a regular star polygon , naturally. Possessing precisely eight edges and eight vertices – a rather predictable count, one might observe. Its identity, for those who appreciate such labels, is formally etched into existence by the Schläfli symbol {8/3}. This isn’t some arcane incantation, just a concise way of stating it’s an 8-pointed star formed by connecting every third vertex of a regular octagon. One could almost call it elegant, if one were inclined towards sentimentality.

For the visually inclined, or those who require exhaustive notation, it can also be denoted as t{4/3} within the realm of Coxeter–Dynkin diagrams . Its inherent balance is reflected in its symmetry group , which is undeniably Dihedral (D 8 ), indicating eight rotational and eight reflectional symmetries. Each internal angle, for those keeping score, measures a precise 45 degrees .

As for its properties, it ticks all the usual boxes for a well-behaved star: it is a star polygon, inherently cyclic (meaning all its vertices lie on a single circumcircle), equilateral (all sides are of equal length), isogonal (all vertices are equivalent), and isotoxal (all edges are equivalent). Its dual polygon ? It’s itself, a rather fitting touch for something so self-contained.

Star Polygons

Humanity seems to have a peculiar fascination with repeating patterns, particularly those that fold back on themselves. The octagram is merely one entry in a surprisingly extensive catalogue of such geometric diversions. Others include:

• The rather ubiquitous pentagram
• The somewhat less common hexagram
• The oddly specific heptagram
• Our current subject, the octagram
• The nine-pointed enneagram
• The ten-pointed decagram
• The eleven-pointed hendecagram
• And finally, the twelve-pointed dodecagram

One might imagine the sheer number of sleepless nights mathematicians have dedicated to these. Or not.

Introduction to the Octagram

In the grand scheme of geometry , an octagram is fundamentally defined as an eight-angled star polygon . This might seem self-evident, but clarity, apparently, is a virtue.

The term “octagram” itself is a rather straightforward linguistic construction, blending a Greek numeral prefix , “octa-” (meaning eight), with the equally Greek suffix “-gram.” This “-gram” suffix, for those who find etymology as thrilling as watching paint dry, is derived from the ancient Greek word γραμμή (grammḗ), which translates quite simply to “line.” [1] So, in essence, an octagram is merely an “eight-line” figure, which, while technically accurate, feels a bit like calling a complex human emotion “a feeling.”

Detail: Unpacking the Octagram

While the regular octagram is often what springs to mind (if anything at all springs to mind regarding octagrams), the term “octagram” in its broader, less restrictive sense, can refer to any self-intersecting octagon . That is to say, any eight-sided polygon where the boundary crosses itself, rather than forming a simple, non-overlapping perimeter. Think of it as an octagon that simply couldn’t stay within its own lines, much like a particularly uninspired artist.

The truly regular octagram, the one that usually adorns geometry textbooks and the occasional esoteric symbol, is precisely designated by the Schläfli symbol {8/3}. This notation isn’t designed to be cryptic, despite appearances. The ‘8’ indicates that it is derived from an 8-sided base polygon (an octagon), and the ‘3’ signifies that its vertices are connected by skipping three points along the perimeter of that underlying octagon. To visualize this, imagine numbering the vertices of a regular octagon from 1 to 8. To draw the {8/3} octagram, you would connect vertex 1 to 4, 4 to 7, 7 to 2, 2 to 5, 5 to 8, 8 to 3, 3 to 6, and finally 6 back to 1. The result is a single, continuous star. It’s a neat trick, if you appreciate efficiency in line drawing.

Variations of the Octagram

Not all octagrams are created equal, which, I suppose, keeps things from becoming entirely monotonous. While the regular octagram exhibits a high degree of symmetry, certain variations exist that possess a reduced dihedral symmetry, specifically Dih 4. This means they might only have four axes of rotational symmetry and four axes of reflectional symmetry, a step down from the regular {8/3}’s eight.

These variations often manifest in different proportions or angles, yielding forms like the “Narrow” or “Wide” octagrams, the latter being a 45-degree rotation of the former, or isotoxal forms where all edges are congruent but the vertices might not be.

Such geometric forms aren’t merely theoretical constructs; they occasionally leak into the real world. For instance, an older iteration of the Flag of Chile once featured an octagonal star geometry, specifically the Guñelve , albeit with its internal edges removed, leaving only the star’s outline. A subtle detail, easily overlooked by those not paying sufficient attention.

Furthermore, the regular octagonal star has found a surprising niche as a recurring symbol for rowing clubs within the Cologne Lowland . One can observe this distinctive emblem proudly displayed on the club flag of the Cologne Rowing Association . It seems even the most mundane of human endeavors find solace in geometric patterns.

The inherent flexibility of octagram geometry also allows for adjustments where, instead of two lines crossing at each intersection, three edges converge at a single central point. This particular configuration is famously embodied by the Auseklis symbol, a Latvian mythological sign.

A more familiar, though perhaps less obvious, manifestation of an octagonal star is the common 8-point compass rose . This navigational aid can be conceptually understood as an octagonal star, delineated by its four primary cardinal points (North, East, South, West) and its four secondary intercardinal points (Northeast, Southeast, Southwest, Northwest). It’s a pragmatic application of star geometry, which is rare enough to be noteworthy.

Even Unicode , the vast repository of digital characters, acknowledges the octagram’s significance. The Rub el Hizb , a prominent Islamic symbol, is represented as a Unicode glyph ۞ at U+06DE. It seems the universe, or at least the digital representation of it, cannot escape these shapes.

As a Quasitruncated Square

The geometric relationship between the simple square and the intricate octagram is more profound than one might initially assume. Through a process of “truncation,” where the corners of a polygon are systematically cut off, one can derive a variety of fascinating forms. Deeper truncations of a square, in particular, can yield isogonal (vertex-transitive) intermediate star polygon forms. These shapes maintain equally spaced vertices but exhibit two distinct edge lengths, adding a layer of controlled complexity.

A straightforward truncation of a square, for example, produces a regular octagon , denoted as t{4}={8}. This is a simple, convex polygon. However, if one ventures into the realm of “quasitruncation” – essentially an inverted truncation – the results are far more intriguing. A quasitruncated square, inverted as {4/3}, directly yields the octagram, t{4/3}={8/3}. [2] It’s a transformation that takes something utterly mundane and twists it into something… well, star-shaped.

This concept extends into three dimensions, because of course it does. The uniform star polyhedron known as the stellated truncated hexahedron , not to be confused with its simpler cousins, is denoted as t’{4,3}=t{4/3,3}. This complex polyhedron features octagram faces, which are directly constructed from the faces of a cube using this very quasitruncation method. For this precise reason, it is often considered a three-dimensional analogue of the octagram, taking the star-like complexity into volumetric space.

To illustrate this geometric progression, consider the following truncations of the square and the cube:

Beyond the stellated truncated hexahedron, another three-dimensional iteration of the octagram is the nonconvex great rhombicuboctahedron , sometimes referred to as the quasirhombicuboctahedron. This particular polyhedron can be conceptualized as a quasicantellated (or quasiexpanded) cube, designated as t 0,2 {4/3,3}. It’s a testament to the endless ways one can manipulate basic forms to achieve increasingly elaborate, and often self-intersecting, structures.

Star Polygon Compounds

For those who find a single star polygon insufficient, the universe obliges with “star polygon compounds.” These are arrangements where multiple star polygons, often identical, are superimposed or interlocked to form a larger, more complex figure.

In the case of octagrammic star figures, there are two primary regular compounds of the form {8/k}. The first, denoted as {8/2}=2{4}, is elegantly constructed from two distinct squares superimposed upon each other, their vertices aligned to form the points of the star. The second, {8/4}=4{2}, is even more degenerate, consisting of four overlapping digons – a digon being a polygon with two sides and two vertices, a concept that stretches the definition of “polygon” to its very limits. Beyond these regular compounds, there exist various other isogonal and isotoxal compounds, including those with rectangular and rhombic symmetries, demonstrating the endless permutations possible when shapes are allowed to intermingle.

The {8/2} compound, or 2{4}, with its Coxeter diagrams represented as + , holds a rather intriguing position in the hierarchy of polyforms. It can be seen as the two-dimensional analogue of higher-dimensional compounds. For instance, it mirrors the three-dimensional compound of cube and octahedron , denoted as + . Extending this pattern, it relates to the four-dimensional compound of a tesseract and a 16-cell, also represented as + , and even the five-dimensional compound of 5-cube and 5-orthoplex . This recurring theme across dimensions highlights the fundamental nature of these geometric relationships, where a n-cube and a cross-polytope find themselves in dual positions, forming a unified, if somewhat convoluted, whole.

Other Presentations of an Octagonal Star

Beyond its formal definitions and compound arrangements, an octagonal star can be conceptualized and presented in several alternative ways, offering different perspectives on its underlying structure.

One can view an octagonal star as a concave hexadecagon – a sixteen-sided polygon – where the internal intersecting lines have been conceptually “erased.” Imagine drawing a regular hexadecagon and then connecting its vertices in a specific, non-sequential order to form the star, and then simply ignoring the inner segments that would complete the hexadecagon. This approach highlights the outer perimeter of the star, emphasizing its overall shape while downplaying its self-intersecting nature.

Alternatively, an octagonal star can be understood through its dissection by radial lines originating from its center. This method breaks down the complex star into a series of simpler triangular or quadrangular segments, each radiating outwards. This perspective is particularly useful for analyzing the star’s internal angles and the relationships between its various points and lines.

Other Uses and Appearances

The octagram, in its various forms, isn’t confined to abstract geometry textbooks. It occasionally surfaces in the most unexpected places, demonstrating humanity’s persistent, if sometimes unconscious, recycling of compelling visual forms.

In the realm of digital typography, Unicode includes the “Eight Spoked Asterisk” symbol, ✳ , designated as U+2733. It’s a simple, elegant representation of an eight-pointed star, ready for deployment in any document or digital interface, should the need for a geometrically precise asterisk ever arise.

Perhaps more compellingly, the distinctive 8-pointed diffraction spikes that so frequently grace the star images captured by the James Webb Space Telescope are a direct consequence of octagram geometry. These spikes aren’t some cosmic anomaly or photographic artifact; they are a predictable result of diffraction caused by the telescope’s unique optical design. Specifically, the hexagonal shape of its primary mirror segments, coupled with the struts that hold the secondary mirror in place, scatter incoming starlight in a way that produces these eight radial lines. It’s a beautiful, albeit entirely physical, manifestation of the octagram, proving that even the most advanced instruments are bound by the predictable laws of light. One might even find it poetic, if one were prone to such flights of fancy. The image of Jupiter ’s moon Europa (on the left) captured by NIRCam notably displays these spikes around the distant stars.

On a slightly less cosmic, and arguably more localized, note, an octagram was notably featured as a parol, or traditional star lantern, for the 2010 ABS-CBN Christmas Station ID. Titled “Ngayong Pasko Magniningning Ang Pilipino” (literally, ‘This Christmas, the Filipinos will Shine’), the choice of an octagonal star was particularly poignant due to its resemblance to the iconic sun symbol found on the Philippine flag . This cleverly imbued the Christmas song with themes of nationalism and patriotism, proving that even geometry can be pressed into service for seasonal propaganda.

See Also

For those whose curiosity remains unsated, or who merely enjoy the exhaustive cross-referencing that Wikipedia so generously provides, further exploration awaits.

• Wikimedia Commons has media related to Octagrams.

Usage

The octagram, in its various guises, has been adopted across cultures and belief systems, a testament to its compelling visual appeal. It seems humans are consistently drawn to patterns that hint at both order and complexity.

• The Rub el Hizb – a prominent Islamic character often used in the Quran.
• The Seljuk star – a significant symbol in Seljuk art and architecture.
• The Shamsa – an intricate medallion or sunburst motif found in Islamic art, often employing octagrammatic patterns.
• The Star of Ishtar – an ancient symbol representing the Sumerian goddess Inanna and her East Semitic counterpart Ishtar , and later associated with the Roman goddess Venus .
• The symbol of Seshat – the hieroglyph representing this ancient Egyptian goddess of wisdom and writing often depicts a seven-petaled flower, which, with its stem, forms a stylized octagram.
• The Star of Lakshmi – an Indian character composed of two overlapping squares, representing the eight forms of wealth (Ashta Lakshmi) associated with the goddess Lakshmi .
• The Surya Majapahit – a solar emblem used during the Majapahit Empire in Indonesia, intended to represent the Hindu gods associated with the cardinal directions.
• The Compass rose – a navigational tool that uses an octagonal star to represent the cardinal directions and the eight principal winds .
• The Auseklis – a powerful symbol in Latvian mythology, often depicted as a regular octagram, associated with the morning star and protection.
• The Guñelve – a representation of Venus in Mapuche iconography, often depicted as an octagram.
• The Selburose – a traditional Norwegian design motif, frequently featuring a regular octagram, often seen in knitting patterns.
• The symbol of Utu – an ancient Mesopotamian god symbol, often depicted as a star, representing the Sun god and justice.
• The Thieves’ Star – a Post-Soviet criminal tattoo symbol, known in Russian as «восьмиконечник», often signifying a “thief in law” or a high-ranking criminal.

Stars Generally

For a broader understanding of stellar geometry, one might delve into:

• Star (polygon)
• Stellated polygons
• Two-dimensional regular polytopes

Others

And for those who simply cannot get enough of polygons with more than eight points, or indeed, other forms of symbolic chaos:

• Hexadecagram
• Dodecagram
• Symbol of Chaos