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Equilateral triangle

An equilateral triangle is a triangle in which all three sides have equal length and all three interior angles measure exactly 60 degrees.^[1]^[2] This geometric figure exhibits the highest degree of symmetry among triangles, featuring three lines of reflection symmetry that bisect each angle and side, as well as rotational symmetries of 120° and 240° around its centroid, in addition to the identity rotation of 0°.^[3] The full symmetry group of an equilateral triangle is the dihedral group D₃, which consists of six elements corresponding to these transformations and preserves distances and angles.^[4] As a special case of an isosceles triangle—where at least two sides are congruent—it is both equilateral and equiangular in Euclidean geometry, making it a regular polygon with three sides.^[5] Key properties include a height of \frac{\sqrt{3}}{2}s (where s is the side length) and an area given by the formula \frac{\sqrt{3}}{4}s^2, derived from dividing the triangle into two 30-60-90 right triangles.^[6] These attributes render the equilateral triangle fundamental in fields such as crystallography, architecture, and advanced mathematics, where its balanced proportions and uniformity are essential.^[7]

Definition and Fundamentals

Definition

An equilateral triangle is a polygon with three sides of equal length and three interior angles each measuring 60 degrees. It is a special case of an isosceles triangle (at least two equal sides) and differs from scalene triangles (all sides of different lengths).^[8] In Euclidean plane geometry, where the sum of the interior angles of any triangle totals 180 degrees, the equality of all three angles follows directly from the equal side lengths, ensuring each is precisely 60 degrees. The term "equilateral" derives from Late Latin aequilateralis, combining aequi- (meaning "equal," from aequus) and lateralis (from latus, meaning "side"), literally translating to "having equal sides."^[9] This nomenclature emphasizes the defining congruence of the sides, typically denoted as all having length a, while the equal angles arise as a necessary consequence in the Euclidean framework, assuming basic familiarity with triangles as three-sided figures formed by straight lines in a plane.^[8]

Basic Characteristics

An equilateral triangle is classified as both equiangular and equilateral, with all three interior angles measuring 60 degrees and all three sides of equal length.^[10] This makes it a special case of an isosceles triangle, where at least two sides are equal, and distinguishes it as the only regular triangle, possessing full rotational symmetry and equal angles and sides. Unlike scalene triangles, which have unequal sides and angles, the equilateral triangle's uniformity ensures that all such elements are congruent.^[10] Due to its high degree of symmetry, all medians, altitudes, angle bisectors, and perpendicular bisectors in an equilateral triangle coincide, intersecting at a single central point that serves as the centroid, orthocenter, circumcenter, and incenter.^[1] This concurrence simplifies many geometric constructions and properties within the triangle. The exterior angles of an equilateral triangle each measure 120 degrees, as each is supplementary to an interior angle of 60 degrees, and their sum totals 360 degrees around the figure.^[11] In non-Euclidean geometries, such as hyperbolic or elliptic spaces, equilateral triangles do not conform to Euclidean characteristics; for instance, their interior angles may sum to less than or more than 180 degrees depending on the curvature, altering the 60-degree equality.^[12]^[13]

Geometric Properties

Sides, Angles, and Symmetry

An equilateral triangle is defined by having all three sides of equal length, denoted as a. This equality implies that the triangle is also equiangular, with all interior angles congruent. The triangle angle sum theorem states that the sum of the interior angles of any triangle is $180^\circ. Therefore, dividing this sum equally among the three angles yields $60^\circ for each: \angle A = \angle B = \angle C = 60^\circ. This property distinguishes the equilateral triangle from other types, ensuring uniformity in both linear and angular measures.^[7] The equilateral triangle exhibits exceptional symmetry, representing the most symmetric two-dimensional polygon with three sides due to its regular structure. Its symmetry group is the dihedral group D_3, which comprises six isometries: three rotations (by $0^\circ, $120^\circ, and $240^\circ about the centroid) and three reflections (across the altitudes from each vertex to the midpoint of the opposite side). These transformations preserve the triangle's shape and size, mapping it onto itself. The rotational symmetries highlight its cyclic order of 3, while the reflections provide mirror invariance along the medians.^[14]) This symmetry extends to isometry properties, where any two equilateral triangles sharing the same side length are congruent via rigid motions—compositions of translations, rotations, and reflections. The side-side-side (SSS) congruence criterion underpins this, as matching all three sides guarantees a rigid motion mapping one triangle precisely onto the other, preserving distances and angles. Such properties underscore the equilateral triangle's role as a fundamental symmetric figure in geometry./01%3A_Teaching_Elementary_Mathematics/1.04%3A_Common_Core_Standards_for_Mathematics/1.4.14%3A_High_School_Geometry_Standards)

Height, Median, and Centroid

In an equilateral triangle with side length a, the height (or altitude) from any vertex to the opposite side is given by the formula h = \frac{\sqrt{3}}{2} a. This length is derived by drawing the altitude, which bisects the base into two segments of length \frac{a}{2} and splits the triangle into two congruent 30-60-90 right triangles, where the side opposite the 30° angle is \frac{a}{2}, the hypotenuse is a, and the side opposite the 60° angle (the height) follows the standard 30-60-90 ratio of $1 : \sqrt{3} : 2, scaled by \frac{a}{2}.^[15] The medians of an equilateral triangle, which connect each vertex to the midpoint of the opposite side, coincide with the altitudes due to the triangle's symmetry, yielding the same length m = \frac{\sqrt{3}}{2} a. All three medians intersect at a single point known as the centroid, which serves as the triangle's center of mass.^[16] The centroid divides each median in a 2:1 ratio, with the longer segment (of length \frac{2}{3} m) directed toward the vertex and the shorter segment (of length \frac{1}{3} m) toward the base midpoint; this property holds for any triangle but is particularly symmetric in the equilateral case.^[17] In an equilateral triangle, the centroid coincides with the orthocenter (intersection of the altitudes) and the circumcenter (center of the circumscribed circle), reflecting the triangle's high degree of symmetry where all principal centers align at one point.^[3]

Area and Perimeter Formulas

The perimeter of an equilateral triangle with side length a is given by the formula P = 3a, which follows directly from the equality of all three sides. The area A of an equilateral triangle can be derived using the base-height formula, where the base is a and the height h = \frac{\sqrt{3}}{2}a splits the triangle into two 30-60-90 right triangles, yielding

A = \frac{1}{2} a h = \frac{1}{2} a \left( \frac{\sqrt{3}}{2} a \right) = \frac{\sqrt{3}}{4} a^2.

This derivation relies on the Pythagorean theorem applied to the right triangle formed by the height.^[18] An alternative derivation uses Heron's formula for the area of a triangle with sides a, a, a. The semiperimeter is s = \frac{3a}{2}, so

A = \sqrt{s(s - a)(s - a)(s - a)} = \sqrt{\frac{3a}{2} \left( \frac{3a}{2} - a \right)^3} = \sqrt{\frac{3a}{2} \left( \frac{a}{2} \right)^3} = \frac{\sqrt{3}}{4} a^2.

This confirms the same expression as the height-based method.^[19] Using trigonometry, the area can also be expressed as half the product of two sides and the sine of the included angle:

A = \frac{1}{2} a \cdot a \cdot \sin 60^\circ = \frac{1}{2} a^2 \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} a^2,

since all angles are $60^\circ.^[18] Among all triangles with a fixed perimeter, the equilateral triangle maximizes the area, a consequence of the isoperimetric inequality for triangles.^[20]

Advanced Properties

Relation to Circles

The incircle of an equilateral triangle with side length a is tangent to all three sides, and its center coincides with the triangle's centroid, orthocenter, and circumcenter. The inradius r, which is the radius of this incircle, is given by r = \frac{\sqrt{3}}{6} a. This formula derives from the height h = \frac{\sqrt{3}}{2} a of the triangle, as the inradius represents the distance from the centroid to any side, equivalent to one-third of the height: r = \frac{h}{3} = \frac{\sqrt{3}}{6} a.^[8]^[21] The circumcircle passes through all three vertices, with its center also at the centroid. The circumradius R is R = \frac{\sqrt{3}}{3} a = \frac{a}{\sqrt{3}}. One derivation uses the fact that the centroid divides each median in a 2:1 ratio, with the longer segment from vertex to centroid being two-thirds of the median length (which equals the height h): R = \frac{2}{3} h = \frac{2}{3} \cdot \frac{\sqrt{3}}{2} a = \frac{\sqrt{3}}{3} a. Alternatively, by the extended law of sines, R = \frac{a}{2 \sin 60^\circ} = \frac{a}{2 \cdot \frac{\sqrt{3}}{2}} = \frac{a}{\sqrt{3}}.^[8]^[22] A distinctive relation in equilateral triangles is r = \frac{R}{2}, which follows directly from substituting the formulas: \frac{\sqrt{3}}{6} a = \frac{1}{2} \cdot \frac{\sqrt{3}}{3} a. This ratio holds uniquely among triangles, underscoring the equilateral's high symmetry.^[8] This coincidence of centers is confirmed by Euler's distance formula between the incenter and circumcenter, d^2 = R(R - 2r). For an equilateral triangle, R - 2r = 0, so d^2 = 0 and d = 0, verifying that the centers are identical.^[23]

Trigonometric and Vector Representations

In an equilateral triangle, all interior angles measure 60°, leading to specific trigonometric values that are fundamental in geometric computations. The sine of 60° is \sqrt{3}/2, the cosine is $1/2, and the tangent is \sqrt{3}. These values derive from the 30°-60°-90° right triangle, formed by drawing an altitude from one vertex to the base of an equilateral triangle with side length 2, which bisects the base into segments of length 1 and the angle into two 30° angles; the altitude length is then \sqrt{3}, yielding \sin(60^\circ) = \opposite/\hypotenuse = \sqrt{3}/2 and \cos(60^\circ) = \adjacent/\hypotenuse = 1/2, with \tan(60^\circ) = \opposite/\adjacent = \sqrt{3}. For coordinate representation in the Cartesian plane, an equilateral triangle with side length 1 can have vertices at (0,0), (1,0), and (0.5, \sqrt{3}/2), placing the base along the x-axis and the third vertex above it.^[24] This configuration scales linearly for a general side length a, with vertices at (0,0), (a,0), and (a/2, a\sqrt{3}/2), facilitating calculations in vector geometry and computer graphics. Vector formulations describe the sides as directed segments of equal magnitude, with 60° angles between consecutive sides at each vertex. Starting from one vertex, the two emanating side vectors \mathbf{u} and \mathbf{v} satisfy |\mathbf{u}| = |\mathbf{v}| and the angle between them is 60°, such that their dot product is \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}|^2 \cos(60^\circ) = |\mathbf{u}|^2 / 2; the vector for the third side, from the end of \mathbf{u} to the end of \mathbf{v}, is then \mathbf{v} - \mathbf{u}. This setup underscores the rotational symmetry, as rotating one vector by 60° around the vertex yields the other. In the complex plane (Argand plane), an equilateral triangle appears prominently in the representation of roots of unity. The non-real cube roots of unity, \omega = e^{2\pi i / 3} = -1/2 + i \sqrt{3}/2 and \omega^2 = e^{-2\pi i / 3} = -1/2 - i \sqrt{3}/2, together with 1, form the vertices of an equilateral triangle inscribed in the unit circle, with side length \sqrt{3} and centroid at the origin. This geometric interpretation aids in analyzing cyclic symmetries and polynomial roots.

Other Mathematical Characteristics

Viviani's theorem states that in an equilateral triangle, the sum of the perpendicular distances from any interior point to the three sides is equal to the altitude of the triangle.^[25] This property holds due to the equal areas of the three smaller triangles formed by connecting the interior point to the vertices, each sharing the same base length as the side of the original triangle and heights equal to the perpendicular distances.^[25] The theorem, originally formulated by Vincenzo Viviani in the 17th century, highlights the uniform distribution of distances in equilateral figures.^[25] The Fermat-Torricelli point of an equilateral triangle, which minimizes the total distance to the three vertices, coincides with the centroid.^[26] In this configuration, the point lies at the intersection of the medians, and the angles subtended by each pair of vertices at this point are all 120 degrees, reflecting the triangle's rotational symmetry.^[26] This coincidence arises because all triangle centers—centroid, orthocenter, circumcenter, and incenter—align in an equilateral triangle.^[26] An equilateral triangle can be dissected into n smaller congruent equilateral triangles only when n = k^2 for some positive integer k, achieved by subdividing each side into k equal segments and connecting the division points parallel to the sides.^[27] For example, with k=2, the triangle divides into 4 smaller ones; with k=3, into 9.^[27] This subdivision preserves orientation and ensures complete coverage without overlaps or gaps.^[27] Among all triangles with a fixed perimeter, the equilateral triangle maximizes the enclosed area, as established by the isoperimetric inequality for triangles.^[28] This follows from Heron's formula, where the area A = \sqrt{s(s-a)(s-b)(s-c)} (with semiperimeter s) is maximized when a = b = c, by the arithmetic mean-geometric mean inequality applied to the side lengths.^[28] The result underscores the equilateral triangle's optimality in balancing perimeter and area constraints.^[28]

Construction Techniques

Compass and Straightedge Method

The classical method for constructing an equilateral triangle using only a compass and straightedge is detailed in Euclid's Elements, Book I, Proposition 1, which provides a foundational construction in Euclidean geometry. This approach begins with a given finite straight line segment as the base and uses circular arcs to locate the third vertex, ensuring all sides are equal.^[29] To perform the construction, first draw the given base segment AB using the straightedge. Place the compass point at A with the radius set to the length of AB, and draw an arc above the line. Next, place the compass point at B with the same radius AB, and draw another arc that intersects the first arc at point C. Connect points A, B, and C with the straightedge to form triangle ABC. This method yields two possible positions for C (one on each side of AB), either of which results in an equilateral triangle.^[30] The proof that triangle ABC is equilateral relies on the properties of circles and congruence. Since point A is the center of the first circle with radius AB, the distance AC equals AB. Similarly, with B as the center of the second circle, BC equals AB. Thus, AB = BC = CA by the side-side-side (SSS) congruence criterion, confirming all angles are 60 degrees. This construction demonstrates the existence of equilateral triangles in the plane, as it logically derives from Euclid's axioms.^[29]^[30] This technique originates from ancient Greek geometry, formalized by Euclid around 300 BCE in his Elements, where it serves as the inaugural proposition to establish basic constructible figures. Equilateral triangles are constructible in this manner because their side lengths involve rational multiples of the base, aligning with the field of constructible numbers generated by quadratic extensions.^[29]^[31] The method requires a pre-given base length AB and cannot directly construct an equilateral triangle from arbitrary angles or without such a segment, limiting its application to scenarios starting from a straight line.^[30]

Coordinate Geometry Approach

One common algebraic method to define an equilateral triangle in the coordinate plane involves placing its base along the x-axis with vertices at (0, 0) and (a, 0), where a > 0 is the side length, and locating the third vertex at \left( \frac{a}{2}, \frac{\sqrt{3}}{2} a \right).^[32] This configuration exploits the triangle's height h = \frac{\sqrt{3}}{2} a, which follows from the Pythagorean theorem applied to the right triangle formed by the base midpoint and the apex.^[32] An alternative approach uses vector rotation to determine the third vertex. Starting from the base vector \vec{v} = (a, 0), rotate it by 60 degrees counterclockwise around the origin using the rotation matrix

\begin{pmatrix} \cos 60^\circ & -\sin 60^\circ \\ \sin 60^\circ & \cos 60^\circ \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}.

Applying this matrix to \vec{v} yields the coordinates \left( \frac{a}{2}, \frac{\sqrt{3}}{2} a \right), confirming the standard placement.^[33] This method aligns with vector representations of the triangle, as explored further in trigonometric contexts.^[33] To verify the equilateral property, apply the Euclidean distance formula to the pairs of vertices. The base distance is |(a, 0) - (0, 0)| = a. The distance from (0, 0) to \left( \frac{a}{2}, \frac{\sqrt{3}}{2} a \right) is

\sqrt{\left( \frac{a}{2} - 0 \right)^2 + \left( \frac{\sqrt{3}}{2} a - 0 \right)^2} = \sqrt{\frac{a^2}{4} + \frac{3a^2}{4}} = \sqrt{a^2} = a,

and similarly for the distance from (a, 0) to the third vertex, ensuring all sides equal a.^[32] This coordinate-based approach offers advantages in computational geometry, enabling efficient algorithmic implementations for tasks such as rendering, collision detection, and geometric transformations in software, where precise numerical computations replace manual constructions.^[34]

Appearances and Uses

In Geometric Figures and Tiling

A regular hexagon can be divided into six equilateral triangles of equal size, with each triangle sharing a common vertex at the hexagon's center and their bases forming the hexagon's sides.^[35] This composition arises because the internal angle of an equilateral triangle is 60 degrees, allowing six such angles to sum to 360 degrees around the center point.^[36] Equilateral triangles also serve as the fundamental units of a triangular grid, which is formed by tiling the plane with these triangles in a regular, repeating pattern where each triangle meets six others at its vertices.^[37] Equilateral triangles tessellate the Euclidean plane without gaps or overlaps in the triangular tiling, a uniform tiling where six triangles meet at each vertex, classified as one of the three regular tilings.^[36] They further appear in the trihexagonal tiling, an Archimedean semiregular tiling variant where equilateral triangles alternate with regular hexagons, with each edge shared between one triangle and one hexagon, and two triangles and two hexagons meeting at every vertex. In related geometric figures, the Reuleaux triangle is constructed from an equilateral triangle by drawing circular arcs centered at each vertex, connecting the other two vertices with radius equal to the side length, resulting in a curve of constant width.^[38] The Koch snowflake begins with an equilateral triangle as its initial shape, upon which smaller equilateral triangles are iteratively added to the midsegments of each side to form a fractal curve.^[39] Similarly, the Sierpinski triangle is generated by starting with an equilateral triangle and recursively removing the central equilateral triangle formed by connecting the midpoints of each side, yielding a self-similar fractal with zero area in the limit.^[40]

In Nature, Art, and Architecture

In nature, the hexagonal structure of beehives constructed by honeybees approximates a tiling based on equilateral triangles, as each hexagon divides into six such triangles, optimizing space and material efficiency for storing honey and rearing brood.^[41] This configuration minimizes wax usage while maximizing strength and volume, a principle proven mathematically superior to alternatives like squares or triangles alone.^[42] Similarly, the crystal lattice of graphene features a honeycomb arrangement of carbon atoms forming equilateral triangles at the atomic scale, contributing to its exceptional strength and conductivity.^[43] In visual arts, equilateral triangles form foundational grids for intricate Islamic geometric patterns, where they interlace with other polygons to create star motifs and tessellations symbolizing infinite unity and divine order.^[44] Celtic knotwork, such as the triquetra, derives from overlapping equilateral triangles to represent interconnected cycles of life, death, and rebirth.^[45] Leonardo da Vinci's Vitruvian Man (c. 1490) incorporates a subtle equilateral triangle between the figure's legs, linking human proportions to biomechanical harmony and the golden ratio, as revealed in recent geometric analysis.^[46] Equilateral triangles appear in architecture for both aesthetic and structural purposes, as seen in the cross-sections of ancient Egyptian pyramids like those at Giza, where the lateral faces approximate equilateral forms to achieve stability and symbolic perfection.^[47] In modern truss designs, such as the Warren truss, equilateral triangular units distribute loads evenly, enhancing rigidity with minimal material in bridges and roofs.^[48] The inverted equilateral triangle shape of yield signs, standardized internationally, ensures immediate visual recognition for traffic control.^[49] Symbolically, the equilateral triangle embodies balance and the trinity across cultures; in Celtic traditions, it signifies the threefold aspects of nature—earth, sea, and sky—evident in motifs like the triquetra.^[50] In Masonic lore, it represents the Deity in emanations of wisdom, strength, and beauty, often enclosing the letter G for Geometry or God.^[51]

Engineering and Scientific Applications

In civil engineering, equilateral triangles are integral to truss bridge designs, particularly in Warren trusses, where they enable uniform load distribution by minimizing force concentrations and optimizing material efficiency under tension and compression. This configuration spreads loads evenly across the structure, making it suitable for long spans with distributed weights. In antenna engineering, equilateral triangular microstrip antennas and dielectric resonator antennas leverage the shape's symmetry to achieve balanced radiation patterns and enhanced signal uniformity, reducing polarization losses in wireless communications.^[52] The inherent rotational symmetry of the equilateral form supports dual-polarization capabilities, improving signal integrity in applications like MIMO systems.^[53] In physics, equilateral triangular arrangements model trigonal planar molecular geometries, such as in boron trifluoride (BF₃), where the central atom bonds to three peripheral atoms at the corners of an equilateral triangle, facilitating symmetric electron distribution and predicting molecular stability via VSEPR theory.^[54] Triangular lattices also exhibit unique wave interference patterns, as seen in Bose-Einstein condensates where density waves emerge from tilted potentials, enabling studies of quantum frustration and topological phases.^[55] In computational algorithms, triangular meshes derived from equilateral triangulations underpin graph theory methods for network analysis, such as Delaunay refinement, which ensures optimal connectivity and bounded element quality for efficient pathfinding and clustering.^[56] In finite element analysis for simulations, equilateral triangular elements provide high accuracy in approximating stress fields and thermoelastic behaviors, as their uniform angles minimize discretization errors in 2D models of material deformation.^[57] Recent advancements in nanotechnology post-2020 highlight equilateral-inspired carbon structures, such as those with Kagome lattices formed by triangle rings, which enhance material ductility and strength for applications in flexible electronics and composites, building on graphene's triangular coordination for superior mechanical resilience.^[58] These designs exploit the lattice's symmetry to achieve high tensile strength while maintaining elasticity, advancing beyond traditional graphene sheets.^[59]

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Nov 11, 2020 · Bees and wasps build space-efficient and strong nests using hexagonal cells.
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What Is It About Bees And Hexagons? : Krulwich Wonders... - NPR
May 14, 2013 · A honeycomb built of hexagons can hold more honey. Maybe hexagons require less building wax. Maybe there's a hidden logic here.
[44]
[2206.14907] Theory of triangulene two-dimensional crystals - arXiv
Jun 29, 2022 · Equilateral triangle-shaped graphene nanoislands with a lateral dimension of n benzene rings are known as [n]triangulenes. Individual [n] ...
[45]
Primary Characteristics of Islamic Geometric Decoration
Most patterns are derived from a grid of polygons such as equilateral triangles, squares, or hexagons. The mathematical term for these grids is "regular ...
[46]
Celtic Knots Explained (And How to Draw One) - Irish Myths
Sep 20, 2024 · Step one: Draw an equilateral triangle with sides that curve slightly outwards. FYI: This is known as a Reuleaux triangle. how to draw a ...
[47]
Hidden Detail in Crotch Solves a 500-Year-Old Leonardo Da Vinci ...
Oct 27, 2025 · Rory Mac Sweeney found a crucial hidden detail, tucked in the Vitruvian Man's crotch: an equilateral triangle that he thinks may explain "one of ...
[48]
[PDF] PYRAMIDS.pdf
Classical pyramids such as the structures at Giza have square bases and lateral sides close in form to equilateral triangles.
[49]
Understanding Trusses: Key Types, Advantages, and Applications
Warren Truss: This truss consists of equilateral triangles. It has an even carrying capability and is generally used for long-span purposes like bridges and ...
[50]
https://www.theirishroadtrip.com/celtic-symbols-and-meanings/
[51]
13 Celtic Symbols and Meanings (Irishman's Guide)
Feb 10, 2025 · The elaborate Triquetra, also known as the Trinity Knot or Celtic Triangle, is one of the most beautiful Irish Celtic symbols and it shows a ...Celtic symbol for mother and son · Celtic Mother Daughter Knot · Irish Symbols
[52]
Encyclopedia Masonica | TRIANGLE - Universal Co-Masonry
The equilateral triangle appears to have been adopted by nearly all the nations of antiquity as a symbol of the Deity, in some of his forms or emanations, and ...
[53]
Equilateral triangular dielectric resonator based co‐radiator MIMO ...
Sep 6, 2018 · The proposed antennas have been designed using an equilateral triangular DR and are excited by a conformal rectangular strip situated at the ...
[54]
Reconfigurable designs of equilateral triangular microstrip antennas ...
In this paper, reconfigurable design of equilateral triangular MSA (ETMSA) loaded with a half U-slot for wideband linear polarized (LP) and dual band dual sense ...
[55]
Triagonal Planar Geometry
Trigonal planar arrangement of 3 regions of high electron density. The regions of high electron density (white) point at the corners of an equilateral triangle.
[56]
Formation of Spontaneous Density-Wave Patterns in dc Driven ...
Apr 19, 2022 · Strongly tilting a Bose-Einstein condensate in a triangular lattice of ultracold atoms induces the formation of pronounced density waves, ...
[57]
[PDF] Delaunay Refinement Algorithms for Triangular Mesh Generation
May 21, 2001 · In theory and practice, meshes produced by. Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and ...
[58]
[PDF] Simulations of thermoelastic triangular cell lattices with bonded ...
Jun 5, 2017 · A finite element model of a two-dimensional equilateral triangular lattice composed of bi-material curved ribs was created by assembling the ...
[59]
A highly ductile carbon material made of triangle rings - AIP Publishing
Jan 24, 2024 · The carbon structure has a Kagome lattice similar to the triangular lattice. Because empirical potentials cannot well simulate mechanical ...Missing: post- | Show results with:post-
[60]
Graphene and Beyond: Recent Advances in Two-Dimensional ...
Focusing on the recent advancements in the field of 2D materials, we discuss theory guided synthesis via different approaches, engineering of atomic defects, ...