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Introduction

Hexagons and octagons are both polygons, but they differ in fundamental ways that influence their geometric properties, structural uses, symbolic meanings, and aesthetic applications. A hexagon has six sides and six angles, while an octagon has eight sides and angles. These simple numeric differences lead to unique characteristics: for example, each interior angle of a regular hexagon is 120°, and each of a regular octagon is 135° . Such distinctions make the hexagon one of only three regular shapes that tile a flat plane without gaps, whereas the octagon cannot tile alone (requiring squares to fill the gaps) . Beyond geometry, these shapes have found very different roles in nature and design. Hexagons famously appear in honeycomb patterns built by bees for optimal strength and space usage, while octagons are seen in man-made symbols like stop signs and in architecture such as eight-sided buildings and domes. Culturally, the hexagon often symbolizes harmony or integration (as in the interlocking triangles of a six-pointed star), whereas the octagon can represent renewal or protection (seen in religious and spiritual iconography). The comparison table below summarizes key properties and examples of each shape:

Aspect	Hexagon (6 sides)	Octagon (8 sides)
Geometry	6 sides; interior angles 120° each (sum 720°) . Symmetric with 6-fold rotation and 6 mirror axes (regular form). Area (regular) = $\tfrac{3\sqrt{3}}{2}a^2$ .	8 sides; interior angles 135° each (sum 1080°) . Symmetric with 8-fold rotation and 8 mirror axes (regular form). Area (regular) = $2(1+\sqrt{2})a^2$ .
Tessellation	Tiles the plane by itself (one of only 3 regular tessellations, along with equilateral triangles and squares) . Three hexagons meet neatly at each vertex (120° × 3 = 360°) .	Cannot tessellate alone (135° × 2 = 270°, ×3 = 405° > 360°). Regular octagons leave gaps; however, they tessellate in a semi-regular pattern when alternated with squares (4-8-8 arrangement around vertices) .
Structural Uses	Ubiquitous in nature: honeycomb cells are hexagonal for maximum area & strength with minimal material . Hexagonal patterns also appear in basalt columns (e.g. Giant’s Causeway) and snowflakes (hexagonal crystal symmetry). Engineering uses hexagon grids for strength-to-weight efficiency (e.g. honeycomb cores in aircraft) .	Rarer in nature; mostly a human-centric shape. Used in architecture and engineering for shape transitions and stability – e.g. octagonal floor plans for towers, domes, and historic structures (many baptisteries, pagodas, and the Dome of the Rock shrine use an octagonal base) . Recognizable in signage (the stop sign’s octagon shape warns drivers) and some structures (e.g. Octagon House designs for efficient interiors).
Symbolism	Often represents harmony, union, and strength in unity. The hexagon’s six points can be seen as two interlocking triangles (upright and inverted) symbolizing the fusion of opposites (spiritual and material) . The honeycomb hexagon further embodies community and cooperation, as many small cells work together for a stronger whole . In sacred geometry, six is a number of balance and cosmic order.	Often symbolizes renewal, transition, and protection. In Christian numerology the octagon (eight) stands for rebirth or a new beginning (the “eighth day” after the seven days of Creation) , hence octagonal baptismal fonts and churches. In Chinese Feng Shui, the octagonal Bagua represents harmony and is used as a protective symbol, uniting the circle (heaven) and square (earth) . Octagons also represent the eight directions and balance in some spiritual frameworks.
Design & Aesthetics	Common in tiling, art, and modern design for a sleek, technical look. Hexagon tiles are popular in flooring and mosaics (their perfect tiling creates visually pleasing, seamless patterns). Many logos and tech designs use hexagons to evoke stability and connectivity (the shape conveys strength and reliability ). In games and graphics, hexagonal grids are favored for their efficient, circular-like adjacency.	An eye-catching shape used when uniqueness is desired. The stop sign is an iconic design using an octagon for instant recognition . Octagonal motifs appear in decorative tiling (often alongside squares) for a classic look (seen in some Victorian floor patterns). In architecture, octagons provide visual interest – e.g. gazebos, domes, and towers with octagonal plans have faceted, almost round aesthetics. The UFC “Octagon” fighting ring and certain brand logos (like umbrella or cross-like designs) use octagonal shapes to stand out.

(Table: Side-by-side comparison of hexagon and octagon properties, uses, and symbolism.)

1. Geometric Characteristics of Hexagon vs Octagon

Sides and Angles: By definition, a hexagon has 6 sides and an octagon has 8. In a regular hexagon (all sides and angles equal), each interior angle is 120°, since the six angles must sum to $ (6-2)\times 180° = 720°$ . In a regular octagon, each interior angle is 135° (sum of $1080°$ for all eight) . The hexagon’s smaller interior angle means it “turns” more sharply at each vertex, whereas the octagon’s larger angle makes it closer to a circle in shape. Both shapes are highly symmetric: a regular hexagon has 6-fold rotational symmetry and 6 reflection axes, while a regular octagon has 8-fold rotational symmetry and 8 reflection axes, reflecting their equal sides and angles.

Area Formulas: The area of a regular polygon can be computed from its side length a. For a regular hexagon, one convenient formula is:

\[ A_{\hexagon} = \frac{3\sqrt{3}}{2}\,a^2, \]

which comes from splitting it into six equilateral triangles . For a regular octagon:

\[ A_{\octagon} = 2(1+\sqrt{2})\,a^2, \]

a formula which can be derived by dividing the octagon into simpler shapes or using trigonometry . In numeric terms, $2(1+\sqrt2)\approx 4.828$, so a regular octagon has about 4.828 times the area of a unit square with the same side length a, whereas a hexagon has $2.598$ times that area (since $3\sqrt3/2 \approx 2.598$). These formulas assume a flat shape; in practical use (like tiling or material usage), the hexagon tends to be very efficient, offering a high area for a given perimeter – a trait it nearly shares with the circle.

Tessellation Behavior: One of the starkest geometric differences is in how these shapes tessellate (tile) a flat surface. A regular hexagon tessellates perfectly by itself: three hexagons meet at each point, their 120° angles adding up to 360°, filling the plane with no gaps . This is famously seen in honeycomb lattices and hexagonal floor tiles. In fact, only three regular polygons can tessellate alone: the equilateral triangle, square, and hexagon . By contrast, a regular octagon cannot tessellate alone; placing octagons edge to edge leaves gaps because 135° × 2 = 270° and 135° × 3 = 405°, overshooting the full 360° around a point. The gaps can be filled by another shape – specifically, a square (90°) can complement two octagons (2×135° = 270°; plus 90° = 360°). This combination yields a well-known semi-regular tiling: an octagon-square checkerboard pattern often used in masonry and tiling . In such a truncated square tiling (denoted 4-8-8), each vertex is surrounded by one square and two octagons . Visually, this pattern looks like an eight-sided star or a lattice of octagons connected by small squares, and it has been used in decorative floor designs for centuries. Regular octagons alone, however, will either overlap or leave holes if you try to pack them on a flat plane .

Illustration of Tiling Differences: The contrast in tiling ability stems from their interior angles. The hexagon’s 120° allows exactly six around a point in a honeycomb fashion (3 hexagons meeting at each vertex, since $3 \times 120° = 360°$) .  Hexagonal honeycomb cells built by honeybees, illustrating the hexagon’s efficient tiling in nature. Bees exploit this property by constructing hexagonal prisms in their comb, which pack perfectly with no wasted space. Octagons, with 135° corners, can only meet in pairs (2×135° = 270°) before exceeding the planar circle; inserting a square (90°) completes the 360° (since $135+135+90 = 360$). This yields the octagon-square tiling seen in some classic tile floors (octagons connected by small square tiles). In summary, hexagons excel at filling space uniformly, whereas octagons need a partner shape to do so.

Other Geometric Traits: Both shapes can be inscribed in circles and circumscribed around circles (they are cyclic polygons). The circumcircle (circle through all vertices) of a regular hexagon has the same radius as the hexagon’s side length , a special property unique to hexagons: effectively, a regular hexagon fits perfectly in a circle with its vertices equally spaced (every 60° around the circle). An octagon’s circumradius is larger than its side (about $1.3065$ times the side length for a regular octagon). The hexagon can also be viewed as six equilateral triangles pieced together , whereas an octagon can be thought of as a square with its corners cut off (indeed, a regular octagon is sometimes described as a truncated square) . This “cutting off” interpretation helps visualize its geometry: starting from a square and slicing off the 4 corners yields an octagon (the removed corners are 45-45-90 triangles). As a result, an octagon is more circular than a square, which is why architects use octagons to transition between a square base and a round dome (more on this in the Structural section).

2. Structural and Engineering Uses

Despite being flat shapes, the hexagon and octagon manifest in very different ways when it comes to structures, engineering, and natural forms. Their inherent geometrical strengths lend themselves to certain applications:

• Hexagon – Nature’s Strongest Pattern: The hexagon is renowned in nature for its structural efficiency. Honeybees construct hexagonal cells in their honeycombs because this shape maximizes storage space while minimizing building material. As noted by observers since antiquity, a hexagonal grid divides a surface into equal regions with the least total perimeter – meaning bees use the least wax to enclose the most honey. Marcus Varro in 36 BC remarked on “the most efficient use of space and building material by the hexagonal shape” , and Charles Darwin later called the honeycomb “absolutely perfect in economizing labor and wax” . Structurally, each cell’s walls meet at 120°, distributing force evenly. This gives honeycombs a high compression strength relative to weight . Engineers have mimicked this: honeycomb sandwich panels – hexagonal cells sandwiched between sheets – are used in aircraft, spacecraft, and vehicle design for lightweight strength . For example, modern aircraft fuselages and wings often contain hexagonal core structures that provide rigidity without heavy solid materials. Even in something like the James Webb Space Telescope, the primary mirror is made of 18 gold-coated hexagonal segments fitted together, forming a roughly hexagonal mirror; this tiling allows a nearly circular, large mirror that folds up compactly. Other natural examples of hexagon geometry include the basalt columnar joints at Giant’s Causeway (as lava cools and contracts, hexagonal cracks form) and the six-fold symmetry of snowflakes (water molecules crystallize in hexagonal lattices). These examples all exploit the hexagon’s tendency to pack efficiently and evenly distribute stress. In summary, the hexagon’s structural mantra is stability through symmetry and efficient packing.
• Octagon – Bridging Shapes in Architecture: Octagons are less common in pure form in nature, but they appear purposefully in architecture and human design. An octagon is often used as a transitional or enclosing shape. For instance, many architectural domes and towers sit on an octagonal base. The Dome of the Rock in Jerusalem is a famous example: an octagonal structure supports a circular dome . The octagon here acts as a bridge between the square building layout and the round dome (an octagon approximates a circle while still using flat, load-bearing walls). Similarly, early Christian baptisteries and some churches were built on octagonal plans – partly for symbolic reasons (as noted, eight signified rebirth), but also for structural reasons: an octagon allows for eight entrances or window facades and spreads the roof load in eight directions, which can be advantageous for domed roofs. In medieval Europe, the Castel del Monte in Italy (13th century) is an outstanding engineering marvel built as a large octagon with octagonal corner towers , demonstrating the shape’s solidity and aesthetic appeal. The castle’s shape provides  eight strong corner angles and an inner courtyard that is also octagonal, yielding a very rigid and evenly balanced structure.
• Stability and Strength: From an engineering perspective, a perfect hexagon is not as inherently rigid as a triangle (which is the most rigid polygon) – hexagons can deform (skew) if not braced, whereas triangles cannot change shape without changing side lengths. That said, in a hexagonal honeycomb each cell is effectively braced by its neighbors, creating a very stiff array. Octagons, being even higher-order polygons, generally require additional support (they might be part of a truss or combined with other shapes). In architecture, octagons often appear in floor plans and sectional profiles of structures rather than as standalone frames. For example, octagon houses were a 19th-century innovation in the United States: proponents like Orson Squire Fowler argued that octagonal homes had more floor space (an octagon encloses ~20% more area than a square of the same perimeter) and allowed better light and airflow from eight directions. A few octagon houses were built, and they tend to have a wraparound verandah and a centralized layout – though the design’s popularity was short-lived, it showcased the octagon’s architectural potential for efficient enclosure. Structurally, an octagon house’s walls meet at 135° which can distribute forces from the roof and wind more evenly around the structure compared to a four-sided house.
• Everyday Examples – Stop Signs and Tools: Perhaps the most ubiquitous engineered octagon is the stop sign. The choice of an octagon for stop signs was very intentional for both recognition and safety. The eight-sided shape is unique among traffic signs: drivers can identify it even from the back (seeing the distinctive outline) and know that oncoming traffic has a stop sign . Early highway engineers also adopted a rule that “the more sides a sign has, the more danger it signifies” – with a circle (infinite sides) for the highest warning (railroad crossing), an octagon for the next level (stop), then diamonds and rectangles for lesser warnings or information . Thus the octagon shape itself conveys urgency and caution, even before one reads the text. This is a clever example of geometric shape influencing behavior. In another domain, octagons show up in the design of some tools and objects: for instance, traditional umbrellas when opened form an octagonal outline; many dumbbell plates and picnic tables are octagonal for practical reasons (dumbbells won’t roll away, tables fit more people). Hexagons, on the other hand, are extremely common in hardware: the heads of bolts and nuts are typically hexagons. A hexagonal bolt head allows a wrench to turn it from 6 different angles, and 6 sides is a good compromise between having many angles for tool grip and having enough flat surface for torque. The hexagon’s 120° angle also means that a wrench needs to turn at most 60° to re-grip the next flat face, which is convenient in tight spaces. This is why the hexagon wrench (Allen key) and hex nuts are standard. Here the hexagon provides functional symmetry.
• Modern Engineering Patterns: Both shapes see advanced engineering use in geodesic structures. Geodesic domes (like Buckminster Fuller’s designs) are composed of triangles, but on a larger scale they project as hexagons and pentagons on a sphere (e.g., a soccer ball has hexagons and pentagons). While octagons don’t feature in geodesic spheres, they do appear in some grids and tiling of 3D space (truncated cubic honeycombs use octagonal prisms). In civil engineering, hexagonal paving stones are sometimes used for driveways because they lock together and resist shifting (plus they can withstand heavy loads when packed). Octagonal paving stones exist too for decorative patterns, but they always pair with squares or rectangles.

In summary, hexagons shine in scenarios requiring efficient, repetitive packing and load distribution (beehives, grids, reinforcement cores, fasteners), whereas octagons are chosen to impart a blend of circular spread with rectilinear ease of construction (buildings, signs, transitional forms). Each shape’s structural use plays to its geometric strengths – the hexagon’s tiling perfection and the octagon’s intermediate geometry between circle and square.

3. Symbolism and Cultural Meanings

Beyond their physical uses, hexagons and octagons carry rich symbolic connotations in various cultures, religions, and systems of thought:

• Harmony of Six (Hexagon): The hexagon often symbolizes integration, balance, and cooperation. This arises partly from the “two triangles” motif: a six-pointed star (the Star of David or hexagram) can be drawn by overlapping an upward and downward triangle, and a regular hexagon lies at the intersection. Thus, the hexagon is associated with the union of opposites – for example, the union of male and female, or of heaven (upward triangle) and earth (downward triangle) . In Hindu and yogic symbolism, a similar six-pointed star (Shatkona) represents the divine union of Shiva (masculine) and Shakti (feminine). Although the star is a separate shape, it underscores the six-ness that the hexagon embodies. The number six itself has ancient numerological meanings: it is often seen as a “perfect” or harmonious number (mathematically, 6 is the sum of its divisors 1+2+3). In Judeo-Christian contexts, God creating the world in six days (and resting on the seventh) gives six a sense of completeness in creation.
  Community and Industry: The beehive’s hexagon has made it a symbol of industry, teamwork, and cooperation. For example, in Napoleon’s time, the honeybee and hexagon were emblems of the French Empire (signifying hard work and unity), and today the state of Utah uses the beehive (and implicitly the hexagon) as its emblem of industriousness. An intact hexagon grid signifies strength through unity – “many parts forming a stable whole.” This is echoed in modern spiritual interpretations: “The hexagon is, at its core, a symbol of integration and harmonious union,” representing how individual pieces join in a greater pattern . The Flower of Life motif in sacred geometry – a pattern of overlapping circles creating a hexagonal lattice of petals – is another example where the hexagon (and its 6-fold symmetry) is thought to represent the interconnection of all life and the underlying geometric order of the universe. Some esoteric traditions associate the hexagon with the heart chakra or with communication between spiritual and material realms, again leaning on the idea of a unifying shape.
  Examples in Iconography: While not as overt as some symbols, hexagons do appear in cultural artifacts. The center of the Star of David is a hexagon, and the star as a whole (with its six outer points) is often interpreted as God’s rule over six directions – north, south, east, west, up, and down . In Buddhism and Eastern art, six-sided mandala patterns sometimes arise (though eight-sided ones are more common for representing cosmology). In contemporary logos, a hexagon shape often conveys technology or science (due to its frequent presence in chemical diagrams and tiling graphics). For instance, the United Nations logo features a world map grid that includes a prominent hexagon pattern in the polar azimuthal projection. Even the floor tiles in mosques or temples might leverage hexagonal geometry as a nod to the interconnectedness of spiritual and earthly realms.
• Power of Eight (Octagon): The octagon’s symbolism frequently centers on renewal, totality, and protection. In early Christian architecture, the octagon was imbued with meaning: eight is one more than seven (the number of days of creation), so it came to signify a new beginning or resurrection. This is why many baptismal fonts and baptistery buildings were octagonal – baptism was seen as a spiritual rebirth, the start of one’s Christian life on the “eighth day” . For example, the Fontana Maggiore in Perugia and the Baptistery of San Giovanni in Florence both have octagonal bases, explicitly to convey the idea of salvation and rebirth. Even some entire churches (like San Vitale in Ravenna) have an octagonal plan, marrying both practical structure and the idea of regeneration.
  Eight Directions and Balance: In Chinese philosophy and Feng Shui, the Bagua is an octagonal diagram with a Yin-Yang symbol in the center and eight trigrams (from the I Ching) around it, each trigram corresponding to a direction and aspect of life (family, wealth, health, etc.). The octagon here is a map of harmonizing forces: it aligns a space or person with all eight facets of their existence. As a Feng Shui tool, octagon-shaped mirrors (Bagua mirrors) are used above doorways to ward off negative energy – the shape is believed to reflect bad influences from all eight directions, offering protection . In Taoist symbolism, the octagon (with the eight trigrams) represents the totality of universal possibilities – all directions in space and all natural elements. We also see the octagon in Islamic culture: the Dome of the Rock (Qubbat al-Sakhrah) is not only an architectural marvel but also carries symbolic meaning by virtue of its shape. Some scholars suggest the octagonal form (with a circle above it) symbolizes the interface of heaven and earth – the dome (heaven) set upon an octagon (perhaps referencing the Islamic view of eight angels supporting the throne of God, or the general idea of paradise having eight gates). In a more secular context, the octagon in stop signs has become a symbol in its own right – universally meaning “STOP.” It’s interesting to note that even without language, an octagonal red sign is immediately understood as a command to halt, showing how a geometric shape gained a worldwide symbolic meaning through convention.
  Spiritual and Mystical Associations: The number eight is often associated with cosmic order or infinity (8 rotated is ∞). In Chinese culture, 8 is a very lucky number (though that’s more to do with pronunciation, the shape plays a role in Feng Shui as mentioned). There are also eight auspicious symbols in Buddhism (the Ashtamangala), but these are not represented by an octagon specifically. However, occasionally the eight-spoked Dharma wheel (symbol of Buddhism) is drawn inside an octagon or circle. In Neo-Pagan and Wiccan traditions, the Wheel of the Year has eight festivals (solstices, equinoxes, and cross-quarters), sometimes represented around an octagonal diagram, linking the shape to the cycle of seasons and renewal. An interesting cross-cultural note: some calendars or systems use base-8 or octagonal symbology (e.g., the medieval ogdoad of primordial deities in Egyptian myth, though abstract, aligns with the concept of eight as a number of completeness before creation).
  Protection and Transition: The octagon’s role as a protective symbol is highlighted by its use in Feng Shui as mentioned (Bagua mirror) and even in folk magic (some stop-sign-esque talismans use octagrams or octagons to protect against evil). Architecturally, an octagonal floor plan gives a sense of enclosure but with more complexity than a square – perhaps why gazebos and pavilions are often octagonal, inviting people to sit at a center with multiple viewpoints of the garden around (symbolically being at the center of the four cardinal and four intercardinal directions). The octagon can thus evoke being at the center of all directions, a balanced vantage point.

In essence, the hexagon’s symbolism gravitates towards unity, harmony, and the strength of interconnected parts, while the octagon’s symbolism touches on renewal, totality (all directions), and spiritual safety or completion. Both shapes have been used deliberately in religious art and architecture to convey these ideas – whether it’s a floor tile pattern imbued with meaning or the very layout of a sacred building.

4. Aesthetic and Design Relevance

Both hexagons and octagons have distinct visual appeals that designers, artists, and architects leverage for aesthetic effect:

• Hexagon in Art & Design: The hexagon’s ability to create a tight, interlocking pattern makes it a favorite in decorative design. Tilework and mosaics frequently employ hexagonal tiles for a clean, modern look (for instance, classic Parisian bathroom floors often use small white hexagon tiles). The seamless tiling of hexagons provides visual continuity – the eye can wander in any direction without hitting a misaligned seam. In graphic design, hexagons are often used to represent technology, science, or digital concepts. A hexagonal grid can suggest a network or honeycomb of information. Many logos incorporate hexagons to convey stability and innovation; for example, some tech companies and even cryptocurrency projects use stylized hexagons, associating the shape with concepts of connectivity, efficiency, and strength . The pattern is also common in backgrounds for brochures or websites related to science: a faint honeycomb pattern immediately gives a “high-tech” or “biological” vibe (since it simultaneously evokes graphene lattices and beehives).
  Artists have also been inspired by hexagons. For instance, the French artist Wassily Kandinsky explored geometric forms and you can find hexagonal motifs in some abstract compositions. In quilting and textiles, the “Grandmother’s Flower Garden” quilt pattern is a classic made of hexagon patches, prized for its beauty and the way hexagons can form flower-like mosaics. Contemporary architects sometimes use hexagonal motifs on facades – the National Aquatics Center in Beijing (Water Cube) uses a bubble-like pattern which is a mixture of hexagons and pentagons reminiscent of a natural foam (a structural nod to hex tiling). Another instance is the exterior of the Musée du Louvre’s Pyramid extension at Lens, which has hexagonal metallic panels. The Metropolitan Museum of Art in New York even had a striking hexagonal tile design on its lobby ceiling (since replaced in renovation, but an example of mid-20th-century use of the shape for a modern aesthetic).
  Visual Appeal: The hexagon’s appeal lies in its combination of order and dynamism. It has symmetry and straight lines like a square, but with an added diagonal slant that gives a sense of movement. Arrange multiple hexagons, and you get a visually satisfying, beeswax-like texture. They can imply continuity beyond the frame of a design (because the pattern can repeat). This is why hexagons are popular in infographic backgrounds and slide designs – they’re interesting but not chaotic. Coloring alternating hexagons can produce mesmerizing optical effects as well. For example, the artist M.C. Escher played with tessellations (though more often triangles and exotic shapes, but the principle extends to hex tessellations).
• Octagon in Art & Design: The octagon is less frequently used as a repeating pattern, but it stands out strongly as an isolated shape or as part of a complex pattern. Stop signs are the most globally recognized use of an octagon in design – their red, bold octagonal form is so iconic that it often finds its way onto T-shirts, posters, and pop art as a symbol (sometimes with the word “STOP” replaced by other messages for parody). In graphic design, an octagon might be used as a background badge or frame (for example, some vintage labels or logos use an octagon outline to give a sense of solidity or heritage). Certain company logos subtly hide octagons; for instance, the logo of the FBI (U.S. Federal Bureau of Investigation) is an inscribed shape that forms an octagon, likely to convey authority and tradition.
  Architecture & Visual Rhythm: Architecturally, octagons create an interesting rhythm. An octagon-based pavilion or gazebo, as mentioned, provides multiple sightlines. Many traditional East Asian pagodas are octagonal at the base or in cross-section – this not only has spiritual meaning (8 directions) but also is pleasing to the eye; the slight faceting as you go around the structure catches light at different angles, giving depth and complexity to the form. In Western classical architecture, octagonal domes or cupolas (like the Florence Baptistery’s dome or the Tower of the Winds in Athens) are admired for their beauty – the octagon shape breaks the monotony of a square and avoids the complete circularity of a cylinder, striking a balance between variety and symmetry.
  In interior design, one finds occasional use of octagonal motifs: e.g., octagon mirrors or tabletops, which tend to look elegant. There was even a trend of octagonal tiles in Victorian times – larger octagon tiles paired with small black squares (the very tessellation described earlier) to make a checker pattern on floors. That style has a classic, vintage appeal and is still used in restorations and retro designs.
  Octagons in Logos and Media: While not as trendy as hexagons in modern logos, octagons do appear. Notably, mixed martial arts (MMA) has “The Octagon” – the eight-sided ring used by the UFC. This has become a brand in itself; the UFC’s use of the octagon conveys toughness and uniqueness. Some sports team logos or event logos might use an octagon to be edgy (literally and figuratively). For instance, the board game Stop Sign or the album art for certain music albums have used octagonal designs for a retro flair.
  Visual Impact: The octagon’s visual impact comes from its rarity and boldness. It’s almost a circle, but not quite – those flat eight sides make it feel constructed and intentional. Thus, an octagon immediately draws attention as something important or special. Designers might choose an octagon when they want to emphasize an element (as traffic engineers did with stop signs to signify importance). In patterns, octagons give a sense of elaboration – you often see them in complex tiling combined with other shapes, which yields a rich, almost tapestry-like geometry (for example, certain Islamic star patterns use an octagonal symmetry underlying them, creating star-octagon tessellations).

In summary, hexagons are beloved in design for their ability to create cohesive patterns and connote modern, networked concepts, whereas octagons are used more sparingly to signal prominence, tradition, or spiritual wholeness. Each shape carries its geometric DNA into the visual language: the hexagon brings a hive of connections, and the octagon brings a halo of significance.

The Dome of the Rock in Jerusalem – an iconic octagonal structure topped with a circular dome. Octagons in architecture often symbolize a union of earth and heaven (square and circle), and provide a visually pleasing, balanced form .

Conclusion

In comparing the hexagon and octagon across geometry, structure, symbolism, and design, we see that each shape has its own “personality.” The hexagon is a master of efficiency – it perfectly fills space, distributes forces evenly, and epitomizes the idea of strength in unity (from bee colonies to carbon nanotubes). Culturally it resonates with themes of harmony and integration, and aesthetically it offers clean, repeating beauty that designers harness for modern appeal. The octagon, on the other hand, is a symbol of transition and completeness – bridging squares to circles in architecture, marking stops and sacred spaces, symbolizing renewal and totality in spiritual contexts. It does not fill space by itself, but perhaps that is fitting: octagons stand alone or with complements to mark something special. Visually, they command attention and convey solidity and balance.

Ultimately, both shapes demonstrate how geometry is far from an abstract concept; it influences everything from how a beehive is built to how we design our cities and sacred structures, and even how we communicate ideas of cosmic order or social warning. Whether it’s the six-sided efficiency of the hexagon or the eight-sided significance of the octagon, these polygons are more than mere shapes – they are foundational patterns woven into the fabric of our natural world and human culture.

Sources: The information in this comparison has been drawn from a variety of references, including mathematical resources (for geometry and tiling properties) , engineering and nature studies (for honeycomb and structural uses) , cultural and historical analyses (for symbolism in Christianity and Feng Shui) , and design insights (for usage in signage and logos) , among others, to ensure a well-rounded and up-to-date comparison. Each shape’s entry in Encyclopedia Britannica, Wikipedia, and specialized texts on sacred geometry and architecture provided detailed context for their unique roles and meanings . This comprehensive look showcases how a simple difference in the number of sides can lead to a wide divergence in functionality and symbolism for these two polygons.

Artists shall rule the future.

Staring at my Prius Lamborghini this morning… The truth of the matter is, it’s probably like the coolest car of all time. The other day I pulled up to curb and there was a little boy who ran and screamed super loud, wow great car!

I think the reason why this was so important to me was, the hilarious thing is this kid who’s like maybe seven years old, probably has no idea about cars in terms of how expensive some are and some are not. Yet he had the personal intelligence of knowing, that my car, with the insanely high visibility 3M wrap, which kind of looks like a combination of hyena meets cheetah meets glitch in the matrix, he obviously knew it was cool.

A lot of people have been asking me about the car, and the truth the matter is, I think the best way to think about it is this like infinite work in progress, it is essentially my personal art car.

The truth is, ever since I was like 15 years old and I got my drivers permit, I’ve actually always been into cars. For me the car has always been a symbol of freedom, pride, performance and more.

What else? To be an American, is all about your car. When you’re out in the streets or in public, nobody has any idea how many bitcoins you own, or how many square foot your house is. Cars are interesting because at the same time, they are insanely superficial, and also, insanely important to society.

For example, Elon Musk who is like probably the greatest innovator and entrepreneur of all time, hit the head on the nail, and knowing that what America wanted and desired and secretly lusted after,,, was like an insanely bold brand new innovative new thought and approached to cars.

The truth is, a car especially a tesla vehicle or any sort of electric car, is probably the worst “investment“, on the planet.  I speak the truth because, I have never spent more than $2500 USD on a car in my whole adult life and I 37 years old. And I’m also the richest person I know, the most wealthy and the most successful person I know, in real life in terms of how poor I grew up, my mom filing for bankruptcy twice, as a result of my dad gambling away the rent money an idea that I learned is, some thoughts like a dozen times, and also, almost becoming homeless twice as well.

I think from an early age, I suppose the upside is that it actually really really taught me some important things about life, money finances, life on the planet etc.

So the first thought is, just do the math. Even as a kid I was able to do this very very simply, essentially… Chelating things from a simple mathematical equation. Like for example if you’re earning $25 an hour as a security guard, you could do the math on how much that $90,000 hellcat is going to run you.

Or even if you’re like a tech employee, making $500,000 a year, but you’re leasing your whole life, and your 2 1/2 kids in your 2 1/2 dogs are all going to private school, and you have outstanding loans from Harvard business school or whatever… And you have some sort of kid you are paying child support for, once again… Do the math, you should know what you can afford and not.

I think the really big idea is, especially after being in Cambodia for like six months and coming back, I think I kind of know the true price of things. And at this point, I’m very reticent to spend money on anything. 

I’ll give you a first example with clothes. There is at least a 10 X markup on clothes, sometimes 50 X to 100 X.

Those cool designer jeans which cost you $200? Maybe a cost like five bucks in Cambodia. Those $700 Balenciaga’s? Maybe seven dollars in a factory in Italy, being made by mainland Chinese people, but in a factory somewhere in the outskirts of Italy.

I think once again cars are like the worst investment of all time because once again… At this point, getting some sort of like secondhand vehicle, a hand me down or some sort of like Prius… Is like by far the best deal on the planet.

For example, I even had this funny idea of a new business idea of just taking old 2010 Priuses, rebranding and re-badging them, and re-upgrading them, with the newest components, maybe throw on some Lamborghini doors, and some gangsta 3M wrap, air edition, and just sell them for a profit as art cars.

In fact, this is what Andy Warhol, and also, Roy Lichtenstein did ,,, for BMW. When you mark something as art, essentially the price on it is infinite.