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Vertical

In geometry and physics, vertical refers to a direction or position that is perpendicular to the horizontal plane, typically aligned with the direction of gravity, such as the line of a plumb bob.^[1] This orientation contrasts with horizontal, which lies parallel to the Earth's surface at a given point, and is fundamental in describing the posture of objects, lines, and forces in both natural and engineered contexts.^[2] For instance, a vertical line on a coordinate plane runs parallel to the y-axis, extending indefinitely upward and downward without intersecting the x-axis.^[3] The concept of verticality has roots in ancient observations of gravity and plumb lines, deriving from the Latin vertex, meaning "highest point," and has evolved to encompass applications across disciplines.^[4] In architecture and engineering, vertical elements like columns or walls provide structural support against gravitational forces, ensuring stability in buildings and bridges.^[5] In physics, vertical motion describes the trajectory of falling objects under acceleration due to gravity, approximately 9.8 m/s² near Earth's surface, as opposed to horizontal motion which remains constant in the absence of friction.^[6] Verticality also plays a key role in biology and ecology, where it influences patterns such as tree growth or ocean layering, with denser water sinking to form vertical gradients in temperature and salinity.^[7] In human anatomy, the term denotes an upright posture, distinguishing bipedal locomotion from horizontal crawling in other species.^[8] These multifaceted uses underscore vertical's importance in defining spatial relationships, from everyday orientations to complex scientific models.

Definition

Geometric Definition

In Euclidean geometry, a vertical direction or line is defined as one that is perpendicular to a designated horizontal plane or line, forming a right angle of $90^\circ. This perpendicularity establishes a fundamental orientation in space, where the vertical serves as the counterpart to the horizontal, enabling the description of positions and alignments relative to a reference base. Such definitions rely on the axioms of Euclidean space, where lines and planes intersect at precise angles without curvature.^[3] A key example of verticality appears in the Cartesian coordinate system, where the y-axis represents the vertical direction in two dimensions, perpendicular to the horizontal x-axis. The origin, at the intersection of these axes, serves as the reference point, with positive y-values extending upward and negative y-values downward. Vertical lines in this system are those parallel to the y-axis, consisting of all points sharing the same x-coordinate, and their equations take the form x = a, where a is a constant. For instance, the line connecting points (3, -2) and (3, 7) is vertical, as it maintains a fixed x-value while varying in y.^[9]^[10] Extending to planes, a vertical plane in Euclidean geometry is one that contains vertical lines or is itself perpendicular to a horizontal plane, thereby including directions aligned with the reference vertical axis. In three-dimensional space, such planes are parallel to the vertical axis (e.g., the yz-plane when the xy-plane is horizontal) and facilitate the projection and intersection analyses central to geometric constructions. Vertical planes maintain parallelism among themselves if they share the same horizontal trace, preserving the 90-degree relation to the base.^[11] To illustrate vertical alignment, consider a simple diagram in the xy-plane: a horizontal baseline along the x-axis from (0, 0) to (4, 0) serves as the reference, with a vertical line rising from (2, 0) to (2, 4), forming a right angle at the base. This setup demonstrates how vertical elements extend orthogonally from horizontal foundations, a principle used in architectural sketches and spatial modeling.^[12]

Physical Definition

In physics, the vertical direction is defined as the orientation aligned with the local force of gravity at a given point, representing the path a freely suspended plumb bob follows due to gravitational pull.^[13] This direction points downward and is opposite to the conventional "up" direction, serving as the reference for local orientation on Earth.^[13] The local zenith marks the point directly overhead along this vertical line, while the nadir indicates the point directly below, both determined by extending the plumb line upward and downward, respectively.^[14] These points establish the vertical axis at any location, with the zenith representing the highest point in the sky relative to the observer and the nadir the lowest.^[14] The gravitational acceleration \vec{g} governs this vertical direction, pointing downward with a magnitude of approximately g \approx 9.81 \, \mathrm{m/s^2}.^[15] In non-uniform gravitational fields, caused by variations in mass distribution, the true vertical—precisely aligned with the local gravity vector—can deviate from the apparent vertical derived from idealized uniform field models, such as a reference ellipsoid.^[13] This physical vertical differs from the purely geometric ideal of perpendicularity to a horizontal plane by incorporating the actual effects of Earth's gravitational field.^[13]

Etymology and History

Etymology

The word "vertical" originates from the Late Latin adjective verticalis, meaning "overhead" or "pertaining to the vertex," derived from vertex, which denotes the highest point, summit, or turning point of something, such as the top of the head or a whirlpool's apex.^[4] This root reflects an association with elevation and direction toward the zenith, contrasting with concepts of flatness or extension. The term entered Middle French as vertical before being adopted into English, carrying connotations of uprightness relative to a reference plane. It was introduced into the English language in the mid-16th century, with the earliest recorded use dating to 1559 in William Cuningham's The Cosmographical Glass, a treatise on cosmography, geography, and astrology that employed the word in astronomical descriptions of overhead positions.^[17] Initially, "vertical" appeared primarily in scholarly contexts involving astronomy, where it described lines or points directly above an observer, and in architectural discussions of upright structures during the Renaissance. By the 1640s, its meaning expanded to generally signify the "highest point" of anything, solidifying its geometric and directional sense.^[4] In comparison, the related term "horizontal" derives from the Greek horizōn kyklos ("bounding circle" or "separating line"), via Late Latin horizon and Middle French horizontal, entering English around 1555 to denote lines parallel to or near the horizon.^[18]^[19] This etymological pairing underscores their perpendicular opposition: vertical lines ascend toward the vertex or zenith, while horizontal ones align with the earth's apparent boundary, a distinction formalized in Renaissance geometry texts, including English translations of Euclid's Elements that adapted classical terms for upright and level orientations.^[20] Early applications in construction, such as aligning plumb lines in building, drew on these geometric roots but evolved into practical standards later.^[4]

Historical Development

In ancient Egypt, the construction of pyramids such as those at Giza involved precise vertical alignments achieved through the use of plumb bobs and sighting techniques tied to astronomical observations. Builders aligned the pyramid bases to cardinal directions by observing circumpolar stars like Kochab in Ursa Minor and Mizar in Ursa Major, which were near the north celestial pole around 2500 BCE, ensuring the structures' sides were perpendicular to the local horizontal plane defined by these sightings.^[21] The internal shafts of the Great Pyramid, inclined but constructed with vertical reference points, were hypothesized to target specific stars at their culminations, demonstrating an integration of vertical measurement with stellar alignment for architectural precision.^[22] Similarly, ancient Greek architects employed vertical concepts in temple construction by aligning structures astronomically. For instance, temples like those at Delphi and Athens were oriented toward solar events.^[23] Vertical plumb lines ensured columns and walls stood true to the local gravity direction during erection. This practice reflected a broader astronomical tradition where sighting rising or setting celestial bodies helped define orientations in building layouts. The formalization of vertical lines in geometry advanced significantly in the 17th century through Girard Desargues' work on perspective. In his 1639 treatise Brouillon project d'une atteinte aux événements des rencontres d'un cône avec un plan, Desargues introduced projective principles that treated vertical lines as parallels converging at infinity in perspective drawings, enabling accurate representation of three-dimensional verticality on a two-dimensional plane for architectural and artistic applications.^[24] During the 18th and 19th centuries, pendulum-based instruments refined the definition of local verticals by aligning with the direction of gravitational acceleration. Christiaan Huygens' pendulum clock of 1656, improved upon in subsequent decades, used the pendulum's swing to establish a precise vertical reference for timekeeping and surveys, as its equilibrium position indicated the local plumb line.^[25] In geodetic efforts, such as the U.S. Coast Survey starting in 1807 and European triangulation networks, pendulums measured variations in gravity to map deflections of the vertical, accounting for the deviation between the plumb line and the geoid normal across large areas.^[26]^[27] In the 20th century, gyroscopes provided inertial references for vertical independent of gravity, with early developments like Elmer Sperry's 1910s gyroscopic stabilizers evolving into artificial horizons for aviation by the 1920s, maintaining a vertical axis via precession.^[28] Satellite geodesy further advanced vertical datums, as seen in the establishment of the North American Vertical Datum of 1988 (NAVD88), which integrated GPS observations with leveling data from the 1929 datum to achieve continent-wide consistency in height references relative to the geoid.^[29] This shift, building on post-World War II satellite launches, enabled global vertical control networks with millimeter accuracy over vast distances.^[30]

Geophysical and Astronomical Contexts

On Earth

On a spherical model of Earth, the local vertical at any surface point aligns with the radial direction toward the planet's center, causing all vertical lines—such as those defined by plumb bobs—to converge in a radial pattern at the geometric center. This idealization assumes uniform density and spherical symmetry, where gravitational attraction acts centrally, making the plumb line a straight path to the core.^[31] Earth's rotation introduces deviations from this pure radial alignment through the centrifugal force, which acts outward perpendicular to the rotation axis. The centrifugal component alters the local vertical by up to approximately 0.1 degrees, most pronounced at mid-latitudes (around 45°) where the meridional component of the centrifugal force maximizes the tilt relative to latitude. This deflection arises because the effective gravity vector—true gravitation minus centrifugal acceleration—tilts slightly from the geocentric radius due to the horizontal component of the centrifugal force, influencing the direction of plumb lines globally.^[32] Geological features further complicate the local vertical, as mass irregularities cause the geoid—the equipotential surface approximating mean sea level—to undulate relative to smoother ellipsoidal models. These geoid undulations, driven by subsurface density variations like mountains or ocean trenches, can reach up to 100 meters in height, leading vertical directions to deviate from idealized radial or ellipsoidal normals. For instance, elevated terrain disrupts local gravity fields, shifting plumb lines away from both geocentric and reference ellipsoid alignments.^[33] The approximate equation for the local vertical deviation due to rotational effects is \delta \approx \frac{\omega^2 R \sin \phi \cos \phi}{g}, where \omega is Earth's angular velocity, R is the planetary radius, \phi is the latitude, and g is gravitational acceleration; this quantifies the centrifugal influence on the tilt in radians.^[34]

In Astronomy

In astronomy, the vertical direction is defined as the line aligned with the local gravitational field on a planetary or stellar surface, extending from the nadir (the point directly below the observer) through the observer to the zenith (the point directly overhead). This alignment ensures that observations account for the local plumb line, which is crucial for accurate pointing of instruments. For instance, in alt-azimuth telescope mounts, the azimuth axis is oriented vertically along this gravitational direction, allowing the telescope to rotate freely in the horizontal plane without balancing requirements, while the altitude axis adjusts the elevation relative to the horizon.^[35] In celestial navigation, vertical circles—great circles on the celestial sphere passing through the zenith and nadir—are used to measure the altitude of stars or other celestial bodies above the local horizon. The altitude is the angular distance along this vertical circle from the horizon to the body, typically ranging from 0° to 90°, and serves as a key input for determining the observer's position by intersecting circles of equal altitude from multiple sightings. This method relies on the vertical being perpendicular to the horizon, defined by the local gravity vector.^[36]^[37] Beyond planetary surfaces, in space environments such as satellite orbits, the vertical direction is defined relative to the central body's gravitational center, approximating the local vertical as the radial vector from the planet's core to the spacecraft. This radial orientation, often termed the local vertical in the Local Vertical Local Horizontal (LVLH) reference frame, is essential for attitude control and orbit maintenance, ensuring instruments or antennas point nadirward or zenithward as needed. For example, satellites requiring radial attitude keep their symmetry axis aligned along this local vertical to maintain stable orientation toward the planet.^[38]^[39] In modern exoplanet studies, the vertical structure of planetary atmospheres is shaped by gravity, which layers the gases through hydrostatic equilibrium, balancing the downward gravitational force against upward pressure gradients. Pressure decreases exponentially with altitude, governed by the relation \frac{dP}{dz} = -\rho g, where P is pressure, z is height, \rho is density, and g is gravitational acceleration, leading to a scale height H = \frac{kT}{mg} that determines atmospheric thickness. This vertical layering influences models of exoplanet climates, such as temperature profiles and chemical distributions, by dictating how radiation and convection interact across pressure levels from the surface to the exosphere.^[40]

Mathematical Formalism

In Two Dimensions

In the Cartesian coordinate system, vertical lines are defined as straight lines parallel to the y-axis, characterized by a constant x-coordinate for all points on the line. These lines are represented by the equation x = c, where c is a real constant that specifies the fixed x-value.^[41]^[42] This form arises because the line extends infinitely in the vertical direction without varying in the horizontal position, making it distinct from lines with a defined slope in the standard y = mx + b form. Vertical lines exhibit several key properties in plane geometry. Their slope is undefined, as the denominator in the slope formula m = \frac{y_2 - y_1}{x_2 - x_1} becomes zero when x_2 = x_1.^[43] All vertical lines are parallel to one another and to the y-axis, meaning they never intersect regardless of their x-values.^[43] Additionally, vertical lines are perpendicular to all horizontal lines, forming right angles of 90 degrees at their intersection points, since horizontal lines have a slope of zero and the product of their slopes is undefined but geometrically confirms orthogonality.^[44] In applications involving graphing and analysis of functions, vertical lines often manifest as asymptotes, indicating points where the function approaches infinity or is undefined. For instance, the rational function f(x) = \frac{1}{x} has a vertical asymptote along the line x = 0, as the denominator equals zero at that point, causing the function values to grow without bound as x approaches zero from either side.^[45] This property is crucial in understanding the behavior of hyperbolic functions and limits in calculus./04:_Day_4/4.03:_Rational_Functions_and_Asymptotes) A practical example of vertical considerations in two-dimensional representations appears in map projections, such as the Mercator projection, where vertical scale distortion becomes pronounced near the poles. In this cylindrical projection, meridians are straight vertical lines, but the increasing scale factor with latitude exaggerates vertical distances, making polar regions appear vastly enlarged compared to equatorial areas.^[46] This distortion preserves angles for navigation but compromises accurate area portrayal in higher latitudes.^[47]

In Three Dimensions

In three-dimensional Euclidean space, the concept of verticality extends from two dimensions by incorporating depth along the z-axis in a standard Cartesian coordinate system, where the xy-plane serves as the horizontal reference. A vertical plane is defined as a plane that contains lines parallel to the vertical direction (the z-axis) and is therefore perpendicular to the horizontal xy-plane.^[48] Such planes can be represented by equations of the form ax + by = d, where the normal vector (a, b, 0) lies in the xy-plane, ensuring no z-component in the normal. Vertical lines in 3D space are straight lines parallel to the z-axis, passing through fixed points in the xy-plane. Their parametric equations take the form x = x_0, y = y_0, z = t, where (x_0, y_0) is a point in the horizontal plane and t varies over the reals, allowing the line to extend infinitely in the vertical direction.^[49] In vector terms, the direction of any vertical line is given by the unit vector \mathbf{k} = (0, 0, 1), which points along the positive z-axis in the standard orientation.^[50] The intersection properties of vertical planes highlight their geometric relationships in 3D. Multiple vertical planes passing through a common point in space all contain the vertical line through that point, forming a pencil of planes—a one-parameter family of planes sharing that fixed vertical axis.^[51] Vertical planes are parallel if and only if their traces (lines of intersection) with the horizontal xy-plane are parallel, in which case they do not intersect at all; otherwise, any two non-parallel vertical planes intersect along a vertical line.^[52] This parallelism ensures that the planes maintain a constant separation in the horizontal direction while extending indefinitely vertically.

Measurement and Applications

Tools for Determining Vertical

The determination of the vertical direction has evolved from rudimentary ancient devices to sophisticated modern instruments, relying fundamentally on the local gravitational field to define the plumb line.^[53] One of the earliest tools, the gnomon—a vertical stick or rod used in ancient observatories—served to establish a reference vertical by aligning with gravity and casting shadows for astronomical observations, as seen in sites like Chaco Canyon dating back over 1,000 years.^[54] The plumb bob remains a foundational gravity-based instrument for precisely identifying the vertical. Consisting of a weight suspended from a string, it aligns itself along the direction of the local gravitational acceleration \mathbf{g}, defining the vertical as the equilibrium position where the tension T in the string balances the weight mg, with m as the mass of the bob and g as the gravitational acceleration.^[55]^[56] This simple device, traceable to ancient Egyptian and Mesopotamian engineering around 3000 BCE, achieves high accuracy in static conditions by minimizing perturbations from air currents or vibrations.^[57] For horizontal alignment, which indirectly infers the vertical as perpendicular to it, the spirit level (or bubble level) employs a vial partially filled with a liquid such as alcohol, where an air bubble seeks the highest point due to buoyancy and the meniscus shape of the liquid surface.^[58] When the bubble centers within etched marks on the vial, the instrument is level, indicating a surface perpendicular to the local vertical; this principle, refined in the 17th century by inventors like Melchisédech Thévenot, allows for precise construction and surveying applications.^[59] In contemporary practice, digital inclinometers utilize micro-electromechanical systems (MEMS) sensors, often accelerometers, to measure tilt angles relative to the vertical with resolutions down to 0.01 degrees.^[60] These devices, integrated into tools for engineering and geotechnical monitoring, detect deviations from the gravity vector electronically, enabling real-time data logging and automation.^[61] Similarly, GPS receivers determine geodetic vertical by computing ellipsoidal heights above a reference datum like WGS84, combined with geoid models to yield orthometric heights accurate to centimeters in differential mode.^[62] This method supports large-scale surveying, as validated in NOAA's height modernization programs.^[63] Laser levels represent another modern advancement for vertical alignment in construction, projecting a visible or invisible beam along the plumb line over distances up to 1,000 feet with self-leveling mechanisms that compensate for setup errors using pendulums or electronic sensors.^[64] These tools ensure plumb walls and alignments in building projects, outperforming manual methods in speed and precision.^[65] The progression culminates in ubiquitous 21st-century applications, such as MEMS accelerometers embedded in smartphones, which measure the device's orientation relative to vertical for features like screen auto-rotation and augmented reality, achieving accuracies sufficient for consumer-level leveling apps.^[60] This integration democratizes vertical determination, building on centuries of refinement from gnomons to digital precision.^[66]

Practical Uses

In architecture and construction, vertical alignment is essential for ensuring structural stability, as load-bearing walls and columns are designed to transfer gravity loads—such as the weight of floors, roofs, and occupants—directly downward to the foundation, minimizing lateral stresses and preventing collapse under compression.^[67] This vertical orientation exploits the natural direction of gravity, allowing materials like reinforced concrete or masonry to efficiently support compressive forces while horizontal elements, such as beams, provide lateral bracing.^[68] For instance, in multi-story buildings, shear walls aligned vertically resist both gravitational and wind-induced loads, enhancing overall rigidity without excessive material use.^[69] In transportation, vertical principles underpin critical mechanics in aviation and vertical transit systems. Vertical takeoff and landing (VTOL) aircraft, such as tiltrotor designs, achieve lift by directing propulsion vertically during hover and ascent, enabling operations in confined urban spaces without runways and transitioning to horizontal flight for efficiency.^[70] This capability relies on aerodynamic principles where vertical thrust counters gravity directly, as seen in configurations like the V-22 Osprey, which balances rotor tilt for stable ascent.^[71] Similarly, elevator systems depend on plumb vertical shafts to guide the cab smoothly along its path, with guide rails aligned to within 1 inch of true vertical per 100 feet to ensure ride quality and prevent misalignment-induced vibrations or derailment.^[72] In arts and media, vertical composition shapes visual storytelling, particularly for mobile-optimized content where portrait orientation aligns with handheld viewing habits. Photographers and filmmakers use vertical framing to emphasize height and subject intimacy, applying rules like the rule of thirds to position key elements along the taller axis, which suits portrait modes on smartphones and enhances engagement on social platforms.^[73] This approach draws the viewer's eye upward, creating a sense of aspiration or focus, as in architectural shots that capture towering structures or cinematic close-ups that fill vertical screens without cropping.^[74] Vertical farming represents an environmental application that optimizes urban agriculture by stacking crops in multi-tiered systems, countering gravity's spatial constraints to produce high yields in limited footprints. These setups, often hydroponic or aeroponic, layer plants vertically to maximize light exposure and nutrient delivery per square meter, reducing land use by up to 90% compared to traditional horizontal fields while enabling year-round production in cities.^[75] Modular designs allow reconfiguration for space efficiency, addressing food security in dense areas by integrating with existing buildings and minimizing transport emissions.^[76]

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The physics of a moving level
A bubble level works the same way as the subway car, with the liquid inside taking the role of the passengers, and the bubble inside making the liquid's motion ...Missing: principle | Show results with:principle
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[PDF] Instruments and methods of physical measurement /
liquid must also be level or horizontal. The spirit level is so con- structed that when the bubble is at its highest point the above condition is fulfilled ...
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Precision Testing for MEMS Accelerometers | NIST
Apr 8, 2016 · They rotate the display on a smartphone from vertical to horizontal, and measure our exercise intensity and activity level in gadgets we carry ...
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Optimizing Reliability of Digital Inclinometer and Flexicurve Ruler ...
After landmarking, the inclinometer was placed on a flat vertical surface and the digital reading was set to zero degrees. The inclinometer was initially ...
[60]
[PDF] Establishing Vertical Control Using GPS Satellite Surveys
If proper procedures are not followed, existing horizontal and vertical distortions, or apparent distortions, in geodetic control networks can cause large ...
[61]
The Vertical Datum: Global Positioning Tutorial
Aug 12, 2024 · The vertical datum is a collection of specific points on the Earth with known heights either above or below mean sea level.
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[PDF] Staying Straight With Rotary Lasers at the Coeur d'Alene Nursery
Figure 3—The Spectra Precision Laser HV401 by Trimble can be used vertically as well as horizontally, is self-leveling, and has a range of more than 1,000 feet.
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Dimensional Tolerances in Construction and for Surface Accessibility
A simple method of establishing the relative elevations of the top and bottom of each ramp is to use a rotating laser level on a tripod and document the ...
[64]
[PDF] A Guide to Smartphone Sensors - Space Math @ NASA
Oct 30, 2018 · The measurement sensitivity of MEMs accelerometers is determined by the ADC resolution and the. ±g range of the analog output. For a ...
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[PDF] Structural Design Loads foe One- and Two- Family Dwellings
This guide is intended to provide a technically sound, concise, and practical method of determining design loads for engineering analysis of residential ...
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[PDF] Design of Load Bearing Wall for Low Rise Building with Partially ...
May 1, 2021 · The mortar and grout in the masonry wall help to stabilize the reinforcement and provide the stability and strength to the wall.
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[PDF] Reinforced Concrete Shear Wall Analysis And Design
Shear walls are vertical structural elements that provide lateral stability to buildings. They are typically made from reinforced concrete and are designed to ...
[68]
[PDF] review of basic principles of v/stol aerodynamics
This paper reviews the principal factors that determine the per- formance o~ V/STOL aircraft. These can be summarized as follows. In hovering, the power ...
[69]
Aerodynamic Characteristics of Propeller-Driven VTOL Aircraft
This paper discusses the two major configurations that are usually considered for achieving VTOL while keeping the fuselage essentially horizontal.
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Elevators for Tall Buildings - Structure Magazine
Elevator contractors generally request, via shop drawing details, the hoistway not vary from plumb by more than 1 inch per 100 feet of vertical travel. Elevator ...Loads · Installation And Access · Industry Advances<|separator|>
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Vertical Video and Using Composition Rules - CSU Social Media Blog
Feb 21, 2023 · For vertical videos and portraits, place the eye of the subject in one of these intersections.
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https://www.structuremag.org/article/elevators-for-tall-buildings/
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Principles of vertical farming - UAF Cooperative Extension Service
Mar 12, 2023 · Growing crops vertically allows for the conservation of space, resulting in a higher crop yield per square foot of land used; this fact is ...<|control11|><|separator|>
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The Future of Farming: Hydroponics — PSCI
Nov 9, 2020 · The modular design of vertical farms allows farmers to alter the layout of the plants to maximize space use and optimize ground space. Since ...