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Reuleaux triangle

The Reuleaux triangle is a two-dimensional curve of constant width formed by the intersection of three circular arcs, each centered at a vertex of an equilateral triangle and passing through the other two vertices, with the width equal to the side length r of the underlying triangle.^[1] This shape, also known as a spherical triangle, maintains a uniform distance between any pair of parallel supporting lines regardless of orientation, distinguishing it from typical polygons.^[2] Named after the 19th-century German mechanical engineer Franz Reuleaux (1829–1905), the figure was formalized in his seminal 1876 treatise The Kinematics of Machinery, where he explored its properties in the context of machine motion and constraint systems.^[3] Reuleaux, often regarded as the father of modern kinematics, used the shape to illustrate concepts like three-point constraints and rotational freedom within fixed boundaries, building on earlier kinematic models dating back to the 1830s in steam engine designs.^[3] His work emphasized the Reuleaux triangle's role in synthesizing mechanisms, influencing engineering education through collections of over 800 physical models preserved at institutions like Cornell University and the Deutsches Museum.^[4] Mathematically, the Reuleaux triangle has the smallest area among all curves of constant width for a given width, calculated as A = \frac{1}{2}(\pi - \sqrt{3})r^2 \approx 0.7048 r^2, which is about 10% less than the area of a circle of the same width.^[1] When rotated within a square of side length r, it traces an envelope that covers approximately 98.77% of the square's area, with the centroid following four elliptical arcs.^[1] These properties arise from its construction, where each arc is a 60-degree segment of a circle of radius r, ensuring rotational symmetry and constant breadth.^[1] Notable applications leverage its constant width and smooth rotation: it forms the basis for square-hole drill bits, which convert circular motion into approximate square paths via an eccentric drive, as demonstrated in 19th-century designs and modern tools.^[5] The rotor in the Wankel rotary engine approximates a Reuleaux triangle, enabling efficient internal combustion within an epitrochoidal housing since its invention in 1957.^[6] Additional uses include steam engine valve controls for precise dwell periods, the constant-width shapes of the British 20p and 50p coins (Reuleaux heptagons) for anti-counterfeiting, and various bearings or cams in machinery requiring uniform motion.^[3]^[7]

History

Origins and early recognition

The Reuleaux triangle, a curve of constant width formed by the intersection of three circular arcs centered at the vertices of an equilateral triangle, appeared in Gothic architecture as early as the late 13th or early 14th century, particularly in ornamental window tracery and structural elements of European cathedrals.^[8] These shapes, characterized by overlapping arcs creating a three-lobed form, were used decoratively in rose windows and niches, as seen in St. Bavo's Cathedral (Sint-Baafskathedraal) in Ghent, Belgium, where Reuleaux-like windows integrated the motif into the intricate stonework of vaulted spaces and porches.^[8] Such applications leveraged the geometric harmony of the form for aesthetic and symbolic purposes, predating its mathematical formalization and reflecting an intuitive grasp of constant-width properties in medieval design.^[8] Around 1500, Leonardo da Vinci sketched figures resembling the Reuleaux triangle in his notebooks, notably in Paris Manuscript A, folio 15v, where a central curved triangular form appears amid mechanical and geometric studies. These drawings suggest da Vinci explored the shape's potential in designs requiring uniform motion or fortification plans, such as rounded bastions that maintain consistent breadth, though he did not explicitly analyze its constant-width attribute. His illustrations, part of broader investigations into gears, cams, and rotational devices, indicate an early practical recognition of the form's utility in engineering contexts. In a paper presented in 1771 and published in 1781 entitled De curvis triangularibus, Leonhard Euler provided the first explicit mathematical discussion of curvilinear triangles and curves of constant width. Euler described such curves, including examples formed by the intersection of three circles centered at the vertices of an equilateral triangle, emphasizing their property of maintaining fixed distance between parallel supporting lines regardless of orientation. This work laid foundational insights into non-circular constant-width shapes, influencing later kinematic studies without yet applying them to specific mechanisms.

Naming and popularization

The Reuleaux triangle was formally introduced by German mechanical engineer Franz Reuleaux in his seminal 1875 work, Theoretische Kinematik (published in English as The Kinematics of Machinery in 1876), where he described it as a curve of constant breadth derived from an equilateral triangle with arcs centered at its vertices. Reuleaux utilized this shape to analyze and synthesize mechanisms capable of approximating straight-line motion, particularly in linkage systems and regulators for steam engines.^[3]^[9] Reuleaux's development of the shape stemmed from his extensive studies of non-circular gears and cams, where the property of constant width—ensuring uniform distance between parallel supporting lines regardless of orientation—proved advantageous for achieving precise, repeatable motions in practical machinery without the variability of circular forms. This emphasis on constant width allowed for more efficient designs in industrial applications, such as control valves and positive-return mechanisms, by minimizing wear and maintaining consistent engagement.^[3]^[9] The conceptual foundations of such shapes can be traced to early 19th-century innovations, including James Watt's steam engine designs, which featured approximate straight-line linkages and planetary gears that indirectly inspired Reuleaux's kinematic explorations. Watt's 1784 parallel motion mechanism, for instance, sought similar goals of linear translation from rotary input, paving the way for Reuleaux's more systematic theoretical framework.^[3]^[9] Reuleaux further popularized the triangle and his kinematic principles through lectures and the display of kinematic models at international expositions, including the 1876 Philadelphia Centennial Exposition, where he served as the German commissioner. These demonstrations, showcasing mechanisms like cams and gears incorporating constant-width curves, captivated engineers and educators, significantly influencing the adoption of kinematics in mechanical engineering curricula across North America and Europe.^[3]^[9]

Definition and construction

Geometric definition

The Reuleaux triangle is defined as the intersection of three circular disks, each of radius w, centered at the vertices of an equilateral triangle with side length w.^[10] This construction ensures the figure is a convex set bounded by a closed curve.^[1] The boundary of the Reuleaux triangle consists of three circular arcs, each subtending a central angle of $60^\circ (or \pi/3 radians) and having radius w, with each arc centered at one vertex of the equilateral triangle and connecting the other two vertices.^[1]^[10] A defining property of the Reuleaux triangle is its constant width: the distance between any pair of parallel supporting lines that touch the boundary remains equal to w, independent of the lines' orientation.^[11] This arises from the symmetric placement of the arcs relative to the equilateral triangle base; for any direction, one supporting line will pass through a vertex, while the parallel line on the opposite side will be tangent to the arc centered at that vertex, maintaining the fixed separation of w due to the $60^\circ arc geometry and the triangle's equal sides.^[11] Visually, the equilateral triangle's vertices act as "anchors" that, combined with the opposing arcs, prevent the width from varying, mimicking the circle's uniformity but with a triangular symmetry.^[1]

Construction methods

The classical method for constructing a Reuleaux triangle using compass and straightedge begins with drawing an equilateral triangle of desired side length w. From each vertex, a circular arc is then drawn with radius w, passing through the other two vertices; the intersection of these arcs forms the curved boundary, replacing the straight sides of the triangle.^[5]^[1] In digital environments, such as CAD software, the Reuleaux triangle is constructed algorithmically by first generating an equilateral triangle and then computing the intersections of three circles, each centered at a vertex with radius equal to the side length w; the boundary is formed by the relevant arc segments between intersection points.^[5]^[12] For manufacturing purposes, the curved boundary is often approximated using polygonal facets or spline curves to facilitate CNC machining or 3D printing. Polygonal approximations involve discretizing each arc into straight-line segments, while spline-based methods, such as uniform cubic B-splines, provide smoother representations with controlled error for practical fabrication.^[13]^[14]

Mathematical properties

Measures of size and shape

The Reuleaux triangle possesses a constant width w, defined as the maximum distance between any two parallel supporting lines, which equals the side length of the base equilateral triangle used in its construction.^[15] This width remains invariant regardless of the orientation of the shape.^[10] The perimeter of the Reuleaux triangle is \pi w, consisting of three circular arcs, each subtending 60 degrees at the opposite vertex and thus collectively spanning 180 degrees of a circle with radius w.^[15] This length matches the perimeter of a circle with the same width, a consequence of Barbier's theorem, which states that all plane convex sets of constant width w have perimeter \pi w.^[15] The area A of the Reuleaux triangle is given by

A = \frac{1}{2} (\pi - \sqrt{3}) w^2 \approx 0.70477 w^2,

computed as the area of the central equilateral triangle plus the areas of three circular segments, each corresponding to a 60-degree sector minus the triangular portion.^[16] For comparison, this area is approximately 89.8% of the area of a circle with the same width w, or \pi w^2 / 4.^[16] The Reuleaux triangle exhibits threefold rotational symmetry about its centroid, which coincides with the centroid of the base equilateral triangle, reflecting its uniform geometric structure.^[17]

Rotational properties

The Reuleaux triangle exhibits threefold rotational symmetry, remaining invariant under rotations of 120° and 240° about its centroid due to its construction from an equilateral triangle with symmetric circular arcs. This symmetry ensures that the shape maps onto itself after these rotations, distinguishing it from curves with higher or lower orders of rotational invariance.^[18] A Reuleaux triangle of constant width w can rotate continuously within a square of side length w, always maintaining contact with the four sides and exhibiting minimal clearance. During this motion, the vertices slide along the square's sides, while the centroid follows a closed path composed of four elliptical arcs with parametric equations

x = 1 + \cos \beta + \frac{\sqrt{3}}{3} \sin \beta, \quad y = 1 + \sin \beta + \frac{\sqrt{3}}{3} \cos \beta

for \beta \in [0, \pi/2] (scaled appropriately for width w).^[1]^[19] The polar moment of inertia about the centroid, derived via integration over the area of the central equilateral triangle and the three circular segments, reflects the distribution of area relative to the centroid, with contributions from the triangle's moment and the segments' shifted moments.^[20] Unlike a circle, the Reuleaux triangle displays a wobbling motion when rolling along a straight line without slipping, pivoting successively at each vertex rather than maintaining continuous tangential contact. The centroid traces an overall straight path parallel to the surface at height w/2, but the instantaneous pivot at a vertex causes the centroid to orbit the contact point at radius w / \sqrt{3}, the distance from centroid to vertex.^[21] For constant linear speed rolling, the angular velocity varies periodically, accelerating during arc contact and decelerating near vertices due to the changing geometry of the contact point relative to the centroid. This variation is evident in the trajectory profiles of Reuleaux-based mechanisms, where maximum angular velocity occurs midway along each arc.^[22]

Extremal and comparative properties

Among all plane curves of constant width w, the Reuleaux triangle achieves the minimal area, given by \frac{1}{2} (\pi - \sqrt{3}) w^2 \approx 0.70477 w^2.^[23] This extremal property is established by the Blaschke-Lebesgue theorem, which proves that no other convex set of constant width can have a smaller area for the same w.^[24] The theorem, originally due to Wilhelm Blaschke and Henri Lebesgue in the early 20th century, has been verified through multiple proofs, including variational and optimal control approaches.^[25] The Reuleaux triangle also exhibits maximal curvature in a discrete sense among constant-width curves, featuring three vertices where the boundary arcs meet at 120° angles.^[26] These vertices introduce points of tangential discontinuity, causing the curvature to jump abruptly from the constant value of $1/w along the arcs to infinite at the corners, marking the sharpest possible corners for any such curve.^[27] This property underscores its role as an extremal case, contrasting with smoother constant-width curves that approximate it but avoid such discontinuities. The optimal packing density of Reuleaux triangles in the plane remains an open problem, with conjectures suggesting values around 0.89 based on numerical simulations of circle packings that approximate the shape.^[28] Compared to the circle, which packs at \pi / \sqrt{12} \approx 0.9069, the Reuleaux triangle's support function h(\theta) exhibits greater variation—ranging between w \left(1 - \frac{1}{\sqrt{3}}\right) \approx 0.423 w and \frac{w}{\sqrt{3}} \approx 0.577 w—reflecting its deviation from circular symmetry while maintaining constant width via h(\theta) + h(\theta + \pi) = w.^[1]

Role as a counterexample

The Reuleaux triangle serves as a prominent counterexample in geometry to the implication that constant width—often termed rotundity—necessitates a differentiable boundary. Although its constant width evokes the smoothness of a circle, the boundary comprises three circular arcs that intersect at the original vertices of the equilateral triangle, creating corners where the tangent direction changes abruptly by 60 degrees, rendering the boundary non-differentiable at those points. This discontinuity arises because the normals to the boundary at each vertex (along the radii from the opposite vertices) form a 60-degree angle, preventing a unique tangent line.^[29]^[1] In metric geometry, the Reuleaux triangle further illustrates that sets of constant width need not possess continuous curvature, even if they are strictly convex. While the set is strictly convex, as its boundary contains no line segments and any line segment joining interior points lies entirely in the interior, the curvature jumps discontinuously at the vertices due to the piecewise circular nature of the boundary. This example underscores that constant width does not require C² smoothness or uniform curvature, distinguishing it from the disk while maintaining the core property of uniform width in all directions.^[30]^[31] The Reuleaux triangle also highlights the failure of the converse to Barbier's theorem, which asserts that every convex plane curve of constant width w has perimeter \pi w. As a convex set of constant width, it satisfies the theorem exactly, yielding the same perimeter as the disk of width w, but the converse—that a curve of perimeter \pi w must have constant width w—does not hold generally, particularly for non-convex shapes where varying width is possible despite fixed perimeter. The Reuleaux triangle thus marks a boundary case, exemplifying the theorem's validity for convex constant-width sets while contrasting with non-convex counterexamples to the converse.^[32]^[15] Historically, Franz Reuleaux employed the triangle to challenge prevailing assumptions in kinematics that equated uniform rolling motion solely with circular forms. In his analysis, he demonstrated that the Reuleaux triangle rolls between parallel lines maintaining constant separation equal to its width, mimicking circular motion in translational aspects but revealing deviations in rotational behavior, such as non-uniform angular velocity. This countered the notion that three single-point contacts suffice to fix a rigid body uniquely, influencing kinematic pair theory and machine design by broadening the scope beyond circular equivalents.^[3]

Applications

Mechanical and industrial uses

The Reuleaux triangle's constant width enables its use in mechanical tools and mechanisms requiring consistent diameter for fitting or motion control, particularly in 19th- and early 20th-century engineering designs. A prominent application is in drill bits for creating approximate square holes. The square hole drill bit, based on a Reuleaux triangle shape, was invented in 1914 by British engineer Harry Watts, who patented a practical version for woodworking and metalworking, constructed from an equilateral triangle with side length s by drawing arcs of radius s centered at each vertex; when rotated in a chuck that permits orbital motion around its centroid, the tool sweeps a path forming a square with filleted corners due to the fixed distance from the centroid to the boundary.^[5] This design produces holes suitable for mortise-and-tenon joints or square drives while avoiding the need for broaching tools.^[5] The rotor in the Wankel rotary engine approximates a Reuleaux triangle, enabling efficient internal combustion within an epitrochoidal housing since its invention in 1957.^[6] In intermittent motion mechanisms, the Reuleaux triangle functions as an eccentric cam to convert continuous rotation into discrete steps with dwell periods. For instance, a Reuleaux triangle cam drives valve motion in steam engines by providing rectilinear reciprocation with pauses, as demonstrated in models from the late 19th-century Reuleaux kinematic collection at Cornell University, which illustrate its role in positive-return systems for precise timing.^[33] These principles extend to early film projectors and clock escapements, where the shape's rotational wobble—arising from non-circular rolling—facilitates stop-motion indexing without slippage, as analyzed in the collection's brass and iron prototypes acquired in 1882 and designated an ASME International Mechanical Engineering Heritage Site in 2004.^[34] The shape also appears in rolling cylinders and bearings for applications demanding uniform contact pressure. Planar bearings incorporating a Reuleaux triangle enable eccentric hypo-cycloid rotation in rotatable shelves, maintaining stable support under load via vertical axis rollers and anti-tipping flanges, as detailed in U.S. Patent 6,568,772 (2003).^[35] Similarly, roller bottles for biological cell cultures use a Reuleaux triangle cross-section to promote translational rolling motion, enhancing gas exchange and medium agitation compared to circular designs, per U.S. Patent 5,866,419 (1999).^[35]

Architectural and design applications

In modern architecture, the Reuleaux triangle has inspired innovative layouts and facades for its unique symmetry and space-filling properties. The Kresge Auditorium at MIT, designed by Eero Saarinen in 1955, features a floor plan outlined by a Reuleaux triangle, topped by a thin-shell concrete dome that one-eighth encloses the spherical segment, optimizing acoustics and enclosure efficiency.^[8] Similarly, antenna arrays at the Arecibo Observatory were optimally arranged along the sides of a Reuleaux triangle to minimize interference in ground screen performance measurements during the 1990s, leveraging the shape's constant width for uniform coverage.^[36] Building facades, such as those in contemporary structures like the Torre Iberdrola in Bilbao, Spain, have incorporated Reuleaux motifs for curved, non-circular symmetry that enhances visual dynamism while maintaining structural balance.^[37] The Reuleaux triangle's constant width has practical applications in consumer product design, enabling smooth handling and stacking. In numismatics, Bermuda issued the world's first Reuleaux triangle-shaped coins in 1996 as part of its Bermuda Triangle commemorative series, with denominations like the $9 silver piece featuring the shape's three-lobed profile for distinctiveness and efficient material use compared to circular coins.^[38] The British 20p and 50p coins also employ a curved equilateral shape of constant width, similar to a Reuleaux triangle, to aid in anti-counterfeiting and vending machine recognition since their introduction in 1982 and 1998, respectively. Guitar picks, such as the American Pick Company's Trident model, adopt the Reuleaux triangle form to provide consistent grip and tonal control, with the curved edges ensuring uniform width for precise string contact.^[39] For utility covers, the shape's property—preventing the lid from falling through its opening due to constant diameter—has been proposed theoretically for manhole covers, though practical implementations remain rare and often limited to valve covers, as seen in examples from San Francisco.^[40] In mechanism design for toys and educational kits, the Reuleaux triangle facilitates demonstrations of rotational symmetry and constant-width motion. Rotating puzzles and gear sets, like the Reuleaux Gear Triangle from i.materialise, use intermeshing Reuleaux forms to create smooth, unidirectional motion without traditional teeth, illustrating kinematic principles in a compact, engaging way.^[41] Educational kits, such as self-balancing Reuleaux triangle devices equipped with gyroscopic sensors, allow users to explore angular momentum and stability, often including customizable firmware for hands-on physics experiments.^[42] These toys highlight the shape's ability to roll like a circle despite its triangular outline, making complex geometry accessible for learning.

Cartographic and symbolic uses

The Reuleaux triangle has been employed in historical cartography for map projections that divide the spherical surface into curved triangular sectors, facilitating representations of the globe with reduced distortion in certain azimuthal views. Notably, Leonardo da Vinci proposed an octant projection in his Codex Atlanticus around 1508, wherein each octant of the sphere is mapped onto a Reuleaux triangle formed by arcs of circles centered at the vertices, allowing for a net-like unfolding of the world map that preserves angular relationships near the poles. This approach minimizes area distortion in polar regions compared to planar projections, making it suitable for early global representations such as da Vinci's world map, which projects the Earth's surface onto eight interconnected Reuleaux triangles. Similarly, English mathematician John Dee utilized a comparable projection in his 1580 polar map, featuring concentric circles within a Reuleaux triangular boundary to depict meridians and parallels with equidistant spacing from the north pole, aiding navigational accuracy in high-latitude contexts. In symbolic and representational contexts, the Reuleaux triangle appears in logos and trademarks due to its constant width, which ensures uniform visibility and scalability without distortion when rotated or resized. For instance, the Colorado School of Mines incorporates a Reuleaux triangle as the core of its legacy emblem, enclosing symbols of mining heritage like a pickaxe, pike, and candle to evoke stability and precision in engineering identity. Military heraldry also adopts the shape; the insignia of the 3rd Marine Logistics Regiment features an escutcheon with a Reuleaux triangle top, symbolizing resilience and threefold unity in operational contexts. Fraternal organizations, such as Mu Beta Psi, use a enameled Reuleaux triangular shield for their crest, representing balanced harmony in musical brotherhood. The Reuleaux triangle serves as a key illustrative tool in educational materials on geometry, particularly for demonstrating curves of constant width. Textbooks and resources often feature diagrams of the shape to highlight its property that the distance between parallel supporting lines remains invariant, contrasting it with circles while showing practical implications like uniform rolling motion. For example, discussions in secondary mathematics curricula, such as those in the National Council of Teachers of Mathematics publications, employ sequences of Reuleaux triangles to teach problem-solving in spatial reasoning and measurement.

Biological and natural occurrences

While the Reuleaux triangle is a precisely defined mathematical figure, exact occurrences in biological or natural contexts are not documented in scientific literature. Approximate triangular forms with curved boundaries appear in some natural structures, such as the segmented skin of the sugar apple fruit (Annona squamosa), which has been described as resembling a Reuleaux triangle due to its rounded, three-lobed shape, though this is a superficial similarity rather than a constant-width curve.^[43] In microscopic organisms, some radiolarian skeletons exhibit geometric symmetry, including triangular elements, potentially providing buoyancy advantages similar to constant-width properties, but these are not true Reuleaux triangles.^[44] The constant width property, however, has inspired biomimetic applications in robotics, such as grippers and locomotion mechanisms that use Reuleaux-like shapes for uniform contact in irregular environments, drawing on evolutionary principles of efficient movement in organisms like microscopic protists for nutrient absorption or navigation.^[45] Rare claims of exact forms in crystal structures or pollen grains exist in popular discussions, but these are debated and generally considered approximations rather than precise matches, with pollen grains typically featuring straight-sided triangular outlines.^[46]

Recent engineering developments

In 2025, researchers developed a frequency-reconfigurable monopole antenna based on a Reuleaux-triangle-shaped radiator, utilizing PIN diodes to switch between ultra-wideband (UWB) and Ku-band operations for satellite communications. This design achieves a wide impedance bandwidth of 3.1–10.6 GHz in UWB mode and 12–18 GHz in Ku-band mode, with peak gains of 4.2 dBi and 5.8 dBi, respectively, leveraging the triangle's constant width for compact, efficient radiation patterns.^[47] Advancements in optical fiber technology emerged in 2024 with the fabrication of Reuleaux triangle core fibers (RTF) exhibiting triple rotational symmetry, enabling enhanced light propagation through chiral twisting. When twisted into a Reuleaux chiral fiber grating (RCFG), these fibers efficiently couple core modes to third-order orbital angular momentum (OAM) vortex modes, supporting applications in high-capacity optical communications with improved mode conversion efficiency exceeding 90%. The structure's equiwidth boundary maintains uniform light guidance, drawing on the inherent rotational symmetry of the Reuleaux triangle.^[48] Sustainable engineering applications saw innovation in 2025 through Venturi Reuleaux triangle (VRA) devices, which generate controlled hydrodynamic cavitation for process intensification in food processing, water treatment, and microbial inactivation. The VRA geometry, with its high perimeter-to-area ratio compared to circular Venturi tubes, is proposed to enhance cavitation intensity while minimizing energy consumption, potentially reducing collapse time and operational costs in theoretical models.^[49] In photocatalysis, porous Reuleaux triangle nanosheets composed of ZnS-CdS-CoSx, developed in 2023 and refined through 2024 studies, have demonstrated superior hydrogen production under visible light. The nanosheets' Z-scheme heterojunction, combined with sulfur vacancies and cobalt sulfide co-catalysts, yields a photocatalytic H2 evolution rate of 18.2 mmol·g⁻¹·h⁻¹, attributed to the triangle's morphology facilitating efficient charge separation and surface area maximization at 128 m²·g⁻¹. This structure outperforms traditional spherical photocatalysts by enhancing light harvesting and stability over 20-hour cycles.^[50] Recent mathematical discoveries in 2024 have extended constant-width shapes beyond the Reuleaux triangle, identifying new families that maintain uniform width in higher dimensions and inspire optimized packing strategies in nanotechnology. These shapes, which generalize the Reuleaux's properties for smoother rolling between parallel surfaces, enable denser arrangements of nanoparticles, potentially improving efficiency in drug delivery and materials synthesis by up to 15% in simulated 3D models.^[51]

Generalizations

Polygonal extensions

A Reuleaux polygon generalizes the Reuleaux triangle to regular polygons with an odd number of sides n \geq 3. It is defined as the intersection of n disks, each of radius equal to the side length of the underlying regular n-gon and centered at its vertices.^[52] This construction ensures the resulting figure is a convex body of constant width equal to the side length.^[52] The boundary of a Reuleaux polygon consists of n circular arcs, each centered at one vertex of the regular n-gon and connecting the vertices at distance equal to the side length.^[53] These arcs form a curvilinear polygon with rotational symmetry of order n. The base case for n=3 yields the Reuleaux triangle.^[52] For n=5, the Reuleaux pentagon maintains constant width but exhibits a larger area than the triangle relative to its width, with the isoperimetric ratio (area to the square of the perimeter) increasing toward that of the enclosing circle as n grows.^[52] This progression highlights how higher-order Reuleaux polygons approximate the circle more closely in terms of efficiency.^[52] Reuleaux polygons are convex only for odd n, as the disk intersection aligns properly to close the boundary without self-intersection.^[52] For even n, the standard construction fails to produce a closed curve of constant width, resulting in spiraling arcs; variants require adjustments such as irregular arc radii or alternative centering to achieve similar properties.^[54]

Three-dimensional analogues

The Reuleaux tetrahedron is the three-dimensional analogue of the Reuleaux triangle, constructed as the intersection of four spheres of radius w centered at the vertices of a regular tetrahedron with edge length w.^[55] This body possesses constant width w in directions from a vertex to the opposite face but exhibits a slightly larger width—varying by up to approximately 2.5%—in directions between pairs of opposite edges, resulting in non-constant diameter across all directions.^[56] The surface consists of four spherical facets meeting along six curved edges formed by the pairwise intersections of the spheres.^[55] The volume of the Reuleaux tetrahedron is given by

V = \left( \frac{8}{3} \pi - \frac{27}{4} \cos^{-1} \left( \frac{1}{3} \right) + \frac{\sqrt{2}}{4} \right) w^3 \approx 0.422 w^3.

^[55] This formula arises from integrating the contributions of the central tetrahedral core and the four spherical segments added at each face, with the exact expression derived using spherical geometry and inclusion-exclusion principles for the ball intersections.^[57] To achieve true constant width in three dimensions, the Meissner tetrahedron modifies the Reuleaux tetrahedron by smoothing its edges.^[58] This construction replaces selected sharp edges—either the three edges meeting at a vertex (yielding one variant) or the three edges surrounding a face (yielding the other noncongruent variant)—with portions of spindle tori generated by rotating circular arcs orthogonal to those edges.^[56] Although often approximated with cylindrical surfaces in basic descriptions, the precise smoothing employs toroidal patches to ensure the supporting planes remain at fixed distance w in all directions, producing a body with four vertices, three unmodified circular edges, four spherical facets, and three toroidal facets.^[58] The resulting Meissner tetrahedra are the conjectured minimizers of volume among all three-dimensional bodies of given constant width w, analogous to the area-minimizing property of the Reuleaux triangle in the plane.^[59] Their volume is

V = \pi \left( \frac{2}{3} - \frac{\sqrt{3}}{4} \cos^{-1} \left( \frac{1}{3} \right) \right) w^3 \approx 0.420 w^3,

computed via decomposition into spherical and toroidal segments.^[56] These three-dimensional constant-width bodies, particularly the Meissner tetrahedron, find applications in mechanical engineering, such as designing rotors that maintain uniform clearance without varying seals, and are readily prototyped via 3D printing to demonstrate rolling behavior akin to spheres.^[60] For instance, 3D-printed models of Reuleaux and Meissner tetrahedra serve as constant-width rotors in educational and experimental setups, extending the rotary principles of two-dimensional Reuleaux shapes used in engines.^[61]

Advanced set-theoretic variants

Yanmouti sets represent a family of convex planar sets that generalize the Reuleaux triangle by varying the radii of the circular arcs centered at the vertices of an equilateral triangle, forming their convex hull. When the arc radii equal the side length of the triangle, the set reduces to the standard Reuleaux triangle; for smaller radii, the sets approach the original equilateral triangle while achieving extremal values in the inequality relating width w, diameter d, and inradius r, specifically w - r \leq d / \sqrt{3}. These sets are of particular interest as they provide the tight bound for this inequality among all planar convex sets, with equality attained in the Reuleaux case.^[62] More broadly, advanced set-theoretic variants of constant-width sets extend beyond symmetric polygonal constructions to include non-symmetric forms generated via the support function h(\theta), where the condition h(\theta) + h(\theta + \pi) = w holds for constant width w.^[63] This formulation allows for irregular boundaries with variable curvature, such as perturbed versions of Reuleaux triangles where the circular arcs are replaced by non-circular curves that satisfy the complementary support condition, enabling asymmetric shapes while preserving constant width. Examples include constructions starting from an arbitrary convex arc over an angular range of \pi and completing the boundary with its 180-degree rotate, yielding sets that deviate from the uniform symmetry of the Reuleaux triangle. These variants are mathematically significant for analyzing support functions, which fully characterize constant-width sets, and for exploring Minkowski sums: the Minkowski sum of two constant-width sets is itself a constant-width set with width equal to the sum of the individual widths, facilitating studies in convex geometry and functional analysis. In optimization contexts, such non-symmetric constant-width sets inform problems like minimizing area under fixed width, where the Reuleaux triangle achieves the global minimum among convex examples per the Blaschke-Lebesgue theorem.

Broader class of constant-width curves

Curves of constant width are convex closed plane curves such that the distance between any pair of parallel supporting lines—known as the caliper diameter or width—is the same in every direction, denoted by w. This property defines a broader class of convex sets beyond circles, encompassing both polygonal and smooth boundaries where the maximum distance between parallel tangents remains invariant under rotation.^[64]^[11] The circle represents the trivial case, with constant width w and support function h(\theta) = w/2 independent of the direction \theta. Non-circular examples include the Reuleaux triangle, which achieves the minimal area among all constant-width curves of given width w by the Blaschke–Lebesgue theorem. Smooth approximations to the Reuleaux triangle, such as those obtained by rounding its vertices while preserving constant width, also belong to this class and can approximate the original shape arbitrarily closely.^[11]^[51]^[27] The study of these curves traces back to Leonhard Euler in 1778, who introduced the concept and termed them orbiform curves while exploring their properties in the context of convex geometry. Subsequent developments by Joseph-Émile Barbier in 1860 advanced the field, notably through his theorem establishing that all curves of constant width w share the same perimeter \pi w, linking them fundamentally to the circle despite their varied shapes.^[65]^[66] Such curves are conveniently parameterized via their support function h(\theta), the signed distance from the origin to the supporting line in direction \theta, satisfying h(\theta) + h(\theta + \pi) = w to ensure constant width. For the circle, h(\theta) is constant at w/2; in contrast, the Reuleaux triangle's support function is the pointwise minimum of the support functions of the three disks of radius w centered at the vertices of the underlying equilateral triangle, reflecting its threefold symmetry and piecewise circular arcs.^[15]^[67]

Key theorems and properties

Barbier's theorem states that every plane curve of constant width w has perimeter \pi w. This result holds regardless of the specific shape of the curve, as long as it maintains constant width, and its proof relies on the Cauchy-Crofton formula from integral geometry, which relates the length of a curve to the measure of lines intersecting it.^[68] The Blaschke-Lebesgue theorem establishes that, among all plane convex sets of constant width w, the Reuleaux triangle encloses the minimal possible area. This theorem highlights the Reuleaux triangle's extremal property in the class of constant-width figures, distinguishing it from other shapes like the circle, which has larger area \pi (w/2)^2 for the same width. The proof involves variational methods showing that any deviation from the Reuleaux form increases the enclosed area.^[69] Tarski's plank problem concerns the minimal total width of planks required to cover a convex body of minimal width w, conjecturing that this total is at least w; the problem was affirmatively solved in the plane by T. Bang in 1951. The Reuleaux triangle provides tight bounds in this context, as its constant width w ensures that coverings by planks of width less than w/2 in certain directions fail to cover the body completely, illustrating the sharpness of the result for minimal-area constant-width sets. Reuleaux's kinematic theorem asserts that a body of constant width w, when rolling without slipping on a straight line, has its center trace a straight path parallel to the line at constant height w/2. This property enables approximate straight-line generation in mechanical linkages and rotors, where the uniform width ensures smooth, non-jerky motion suitable for engineering applications like uniform rotation mechanisms.^[68]

Other geometric figures of interest

The Reuleaux triangle is constructed by taking an equilateral triangle as its central "skeleton" and replacing each side with a 60-degree circular arc centered at the opposite vertex, using the side length w as the radius.^[1] The area of this underlying equilateral triangle is \frac{\sqrt{3}}{4} w^2, which is smaller than the Reuleaux triangle's area of \frac{1}{2} (\pi - \sqrt{3}) w^2 \approx 0.704 w^2; this contrast highlights how the Reuleaux figure expands the base shape while achieving the minimal area possible for any plane curve of constant width w, as established by the Blaschke–Lebesgue theorem.^[1] In the broader family of Reuleaux polygons, which generalize the triangle to odd numbers of sides, the circle emerges as the limiting case as the number of sides increases indefinitely.^[53] Unlike the Reuleaux triangle's intermittent pivoting during rolling, the circle enables smooth, uniform rotation without such discontinuities, serving as the ideal constant-width shape for applications requiring perfect isotropy.^[53] The Reuleaux triangle also connects to rotors in mechanisms, where it rotates fully within a square of side w, maintaining contact with all four sides and sweeping nearly the entire square's area—an property exploited in 19th-century tools like the Watt drill for approximating square holes.^[1] This rotor behavior facilitates tiled approximations of squares in engineering contexts, where multiple Reuleaux shapes can interlock to mimic rectangular profiles.^[1] Finally, the paths traced by fixed points on a rotating or rolling Reuleaux triangle generate roulette curves, linking the figure to classical geometry through families of cycloidal and trochoidal traces studied since antiquity.^[70] These roulettes, including variants akin to the hippopede formed by circular motions, underscore the Reuleaux triangle's role in exploring intersections between constant-width sets and dynamic curve generation.^[70]

References

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### Summary of Reuleaux Triangle Applications in Kinematic Mechanisms
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