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75-427: Types of stairs include staircases (also called stairways) and escalators . Some alternatives to stairs are elevators (also called lifts), stairlifts , inclined moving walkways , ladders , and ramps. A stairwell is a vertical shaft or opening that contains a staircase. A flight (of stairs) is an inclined part of a staircase consisting of steps (and their lateral supports if supports are separate from steps). This

150-570: A φ {\displaystyle \;r(\varphi )=a\varphi \;} one gets the conical spiral (see diagram) Any cylindrical map projection can be used as the basis for a spherical spiral : draw a straight line on the map and find its inverse projection on the sphere, a kind of spherical curve . One of the most basic families of spherical spirals is the Clelia curves , which project to straight lines on an equirectangular projection . These are curves for which longitude and colatitude are in

225-462: A ( φ 2 + 1 ) 3 / 2 {\displaystyle \kappa ={\tfrac {\varphi ^{2}+2}{a(\varphi ^{2}+1)^{3/2}}}} . Only for − 1 < n < 0 {\displaystyle -1<n<0} the spiral has an inflection point . The curvature of a logarithmic spiral r = a e k φ {\displaystyle \;r=ae^{k\varphi }\;}

300-404: A ( arctan ⁡ ( k φ ) + π / 2 ) {\displaystyle \;r=a(\arctan(k\varphi )+\pi /2)\;} and k = 0.2 , a = 2 , − ∞ < φ < ∞ {\displaystyle \;k=0.2,a=2,\;-\infty <\varphi <\infty \;} one gets a spiral, that approaches

375-438: A degenerate case (the function not being strictly monotonic, but rather constant ). In x {\displaystyle x} - y {\displaystyle y} -coordinates the curve has the parametric representation: Some of the most important sorts of two-dimensional spirals include: An Archimedean spiral is, for example, generated while coiling a carpet. A hyperbolic spiral appears as image of

450-483: A logarithmic spiral r = a e k φ {\displaystyle \;r=ae^{k\varphi }\;} is   L = k 2 + 1 k ( r ( φ 2 ) − r ( φ 1 ) )   . {\displaystyle \ L={\tfrac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}\ .} The inversion at

525-453: A logarithmic spiral r = a e k φ {\displaystyle \;r=ae^{k\varphi }\;} is   A = r ( φ 2 ) 2 − r ( φ 1 ) 2 ) 4 k   . {\displaystyle \ A={\tfrac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2})}{4k}}\ .} The length of an arc of

600-487: A spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects . Two major definitions of "spiral" in the American Heritage Dictionary are: The first definition describes a planar curve, that extends in both of the perpendicular directions within its plane;

675-462: A 90° angle landing. Stairs may also return onto themselves with 180° angle landings at each end of straight flights forming a vertical stairway commonly used in multistory and highrise buildings. Many variations of geometrical stairs may be formed of circular, elliptical and irregular constructions. Ascending a set of stairs to a higher floor is often referred to as going "upstairs", the opposite being "downstairs". The same words can also be used to mean

750-454: A circular motion around it is a " helix ". Since the very purpose of a stairway is to change elevation, it is inherently a 3-dimensional path. Loose everyday usage conflates the terms helical and spiral , but the vast majority of circular stairs are actually helical. True spiral staircases would be nonfunctional flat structures, although functional hybrid helical spiral staircases can be constructed. This article attempts to preferentially use

825-459: A closed curve around a fixed axis: the shape of the curve remains fixed, but its size grows in a geometric progression . In some shells, such as Nautilus and ammonites , the generating curve revolves in a plane perpendicular to the axis and the shell will form a planar discoid shape. In others it follows a skew path forming a helico -spiral pattern. Thompson also studied spirals occurring in horns , teeth , claws and plants . A model for

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900-502: A corner of the square). A "pure helix" fills a circular stairwell, and has multiple steps and handrail elements which are identical and positioned screw-symmetrically . Helical stairs have a handedness or chirality , analogous to the handedness of screw threads , either right-handed or left-handed helical shapes. Ascending a right-handed helix rises counter-clockwise, while ascending a left-handed helix rises clockwise (both as viewed from above). A fundamental advantage of helical stairs

975-423: A curve with polar equation r = r ( φ ) {\displaystyle r=r(\varphi )} is For the spiral r = a φ n {\displaystyle r=a\varphi ^{n}\;} the length is Not all these integrals can be solved by a suitable table. In case of a Fermat's spiral, the integral can be expressed by elliptic integrals only. The arc length of

1050-534: A helix with a special central projection (see diagram). A hyperbolic spiral is some times called reciproke spiral, because it is the image of an Archimedean spiral with a circle-inversion (see below). The name logarithmic spiral is due to the equation φ = 1 k ⋅ ln ⁡ r a {\displaystyle \varphi ={\tfrac {1}{k}}\cdot \ln {\tfrac {r}{a}}} . Approximations of this are found in nature. Spirals which do not fit into this scheme of

1125-552: A linear relationship, analogous to Archimedean spirals in the plane; under the azimuthal equidistant projection a Clelia curve projects to a planar Archimedean spiral. If one represents a unit sphere by spherical coordinates then setting the linear dependency φ = c θ {\displaystyle \varphi =c\theta } for the angle coordinates gives a parametric curve in terms of parameter ⁠ θ {\displaystyle \theta } ⁠ , Another family of spherical spirals

1200-533: A loxodrome projects to a logarithmic spiral in the plane. The study of spirals in nature has a long history. Christopher Wren observed that many shells form a logarithmic spiral ; Jan Swammerdam observed the common mathematical characteristics of a wide range of shells from Helix to Spirula ; and Henry Nottidge Moseley described the mathematics of univalve shells. D’Arcy Wentworth Thompson 's On Growth and Form gives extensive treatment to these spirals. He describes how shells are formed by rotating

1275-530: A much stronger domestic and status role than a military function" and that "there are sufficient examples of anticlockwise stairs in Britain and France in [the 11th and 12th centuries] to indicate that the choice must have depended both on physical convenience and architectural practicalities and there was no military ideology that demanded clockwise staircases in the cause of fighting efficiency or advantage". Developments in manufacturing and design have also led to

1350-450: A multi-story building, and are a functional part of it. Stairwells are often used to place several stairs one above the other to save vertical space. The entrance to elevators is often located in the stairwell. Sometimes the stairwell goes around the lift shaft , other times it is placed next to it. An exterior stairway is a stair in a separate structure attached to the rest of the building body, and can either be enclosed or exposed to

1425-408: A safe forward-facing descent of very steep stairs (however, designs with recessed treads or footholds do not have this feature). The treads are designed such that they alternate between treads for each foot: one step is wide on the left side; the next step is wide on the right side. There is insufficient space on the narrow portion of the step for the other foot to stand, hence the person must always use

1500-423: A shape similar to that of the letter "Z" if seen from above. The use of landings and a possible change of direction have the following effects: Other forms include stairs with winders that curve or bend at an acute angle, three flights of stairs that join at a landing to form a T-shape, and stairs with balconies and complex designs. A "mono string" staircase is a term used for a staircase with treads arranged along

1575-449: A single steel beam. A "double string" staircase has two steel beams, one on either side, and treads spanning between. The term "helical stair" has many synonyms: The helical stair is also called the spiral stair, winding stair, circular stair, elliptical stair, oval stair, geometric stair, vis, vice, vis de Saint Gilles, St. Gilles screw, belfry stair, turret stair, caracole, turnpike, cochlea, cockle, corkscrew, and ascensorium. Helical

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1650-464: A stair, in particular the rise height and going of the steps, should remain the same along the stairs. The following stair dimensions are important: Stairs can take a large number of forms, combining straight runs, winders, and landings. The simplest form is the straight flight of stairs, with neither winders nor landings. These types of stairs were commonly used in traditional homes, as they are relatively easy to build and only need to be connected at

1725-494: Is κ = 1 r 1 + k 2 . {\displaystyle \;\kappa ={\tfrac {1}{r{\sqrt {1+k^{2}}}}}\;.} The area of a sector of a curve (see diagram) with polar equation r = r ( φ ) {\displaystyle r=r(\varphi )} is For a spiral with equation r = a φ n {\displaystyle r=a\varphi ^{n}\;} one gets The formula for

1800-425: Is In case of an Archimedean spiral ( n = 1 {\displaystyle n=1} ) the polar slope is tan ⁡ α = 1 φ   . {\displaystyle \;\tan \alpha ={\tfrac {1}{\varphi }}\ .} In a logarithmic spiral ,   tan ⁡ α = k   {\displaystyle \ \tan \alpha =k\ }

1875-427: Is a common misconception that helical staircases in castles rose in a clockwise direction, to hinder right-handed attackers. While clockwise helical staircases are more common in castles than anti-clockwise, they were even more common in medieval structures without a military role, such as religious buildings. Studies of helical stairs in castles have concluded that "the role and position of spirals in castles ... had

1950-465: Is a handrail at both sides of the treads. These designs have the advantage of a more uniform tread depth when compared to a narrow helical staircase. Such stairs may also be built around an elliptical or oval footprint, or even a triangular or pentagonal core. Lacking a central pole, an open well staircase is supported at its outer periphery, or in some cases may be a completely self-supporting and free-standing structure. An example of perimeter support

2025-512: Is also the name given to spiral shaped fingerprints . A spiral like form has been found in Mezine , Ukraine , as part of a decorative object dated to 10,000 BCE. Spiral and triple spiral motifs served as Neolithic symbols in Europe ( Megalithic Temples of Malta ). The Celtic triple-spiral is in fact a pre-Celtic symbol. It is carved into the rock of a stone lozenge near the main entrance of

2100-498: Is an excerpt from Staircase . The concept of stairs is believed to be 8000 years old, and are one of the oldest structures in architectural history. The oldest example of spiral stairs dates back to the 400s BC . Medieval architecture saw experimentation with many different shapes, and the Renaissance even more so with varied designs. A stair , or a stairstep , is one step in a flight of stairs. A staircase or stairway

2175-501: Is constant. The curvature κ {\displaystyle \kappa } of a curve with polar equation r = r ( φ ) {\displaystyle r=r(\varphi )} is For a spiral with r = a φ n {\displaystyle r=a\varphi ^{n}} one gets In case of n = 1 {\displaystyle n=1} (Archimedean spiral) κ = φ 2 + 2

2250-416: Is easier to design and construct than one with landings or winders. The rhythm of stepping is not interrupted in a straight run, which may offset an increased fall risk by helping to prevent a misstep in the first place. However, many long straight runs of stairs will require landings or winders to comply with safety standards in building regulations. Straight stairs can have a mid-landing incorporated, but it

2325-435: Is not unusual to find a skylight or roof windows above a stairwell. A research article suggests that perceived safety increases when using downwards lighting towards the stairs, while upwards lighting resulted in a substantial decrease of perceived safety. In new stairways, some jurisdictions require handrails to be mounted on both sides. Often the stairs in a stairwell are an emergency exit in case of fire, and

Stairs - Misplaced Pages Continue

2400-422: Is one or more flights of stairs leading from one floor to another, and includes landings, newel posts, handrails, balustrades , and additional parts. In buildings, stairs is a term applied to a complete flight of steps between two floors. A stair flight is a run of stairs or steps between landings. A stairwell is a compartment extending vertically through a building in which stairs are placed. A stair hall

2475-408: Is possibly the least poetic but also the most accurate generic title. The term " spiral " has a more narrow definition in a mathematical context, as a curve which lies in a single plane and moves towards or away from a central point, with a continuously changing radius. The mathematical term for the 3-dimensional curve traced where the locus progresses at a fixed radius from a fixed line while moving in

2550-525: Is probably more common to see stairs that use a landing or winder to produce a bend in the stairs. A straight flight with a mid-landing will require a lot of straight length, and may be more commonly found in large commercial buildings. L-shaped stairways have one landing and usually a change in direction by 90°. U-shaped stairs may employ a single wider landing for a change in direction of 180°, or two landings for two changes in direction of 90° each. A Z-shaped staircase incorporates two opposite 90° turns, creating

2625-555: Is that they can be very compact, fitting into very narrow spaces and occupying a small footprint. For this reason, they can often be found in ships and submarines , industrial installations, small loft apartments , and other locations where floorspace is scarce. However, this compactness can come at the expense of requiring great craftsmanship and care to produce a safe and effective structure. By contrast, grand helical stairs occupying wide sweeps of space can also be built, to showcase luxurious funding and elegant taste. Architects have used

2700-456: Is the rhumb lines or loxodromes, that project to straight lines on the Mercator projection . These are the trajectories traced by a ship traveling with constant bearing . Any loxodrome (except for the meridians and parallels) spirals infinitely around either pole, closer and closer each time, unlike a Clelia curve which maintains uniform spacing in colatitude. Under stereographic projection ,

2775-573: Is the Vatican stairwell or the Gothic stairwell. The latter stairwell is tight because of its location where the diameter must be small. Many helices, however, have sufficient width for normal-size treads (8 inches (200 mm)) by being supported by any combination of a center pole, perimeter supports attaching to or beneath the treads, and a helical handrail. In this manner, the treads may be wide enough to accommodate low rises. In self-supporting stairs,

2850-458: Is the stairs, landings, hallways, or other portions of the public hall through which it is necessary to pass when going from the entrance floor to the other floors of a building. Box stairs are stairs built between walls, usually with no support except the wall strings. Stairs may be in a "straight run", leading from one floor to another without a turn or change in direction. Stairs may change direction, commonly by two straight flights connected at

2925-738: The x {\displaystyle x} - y {\displaystyle y} -plane a spiral with parametric representation is given, then there can be added a third coordinate z ( φ ) {\displaystyle z(\varphi )} , such that the now space curve lies on the cone with equation m ( x 2 + y 2 ) = ( z − z 0 ) 2   ,   m > 0 {\displaystyle \;m(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;} : Spirals based on this procedure are called conical spirals . Starting with an archimedean spiral r ( φ ) =

3000-448: The Renaissance even more so with varied designs. Good lighting is important in a staircase so users see where they are going and to prevent falls. There is often a window on the wall to let in daylight . In many cases, indoor stairs are placed far inside the building structure, and it is often not easy to get access to a wall on the outside where it would be natural to have a regular window for letting daylight in. For this reason, it

3075-720: The cella . The temple was constructed around 480–470 BCE. When used in Roman architecture , helical stairs were generally restricted to elite luxury structures. They were then adopted into Christian ecclesiastic architecture . During the Renaissance and Baroque periods, increasingly spectacular helical stairways were devised, first deleting walled enclosures, and then deleting a central post to leave an open well. Modern designs have trended towards minimalism, culminating in helical stairs made largely of transparent glass, or consisting only of stair treads with minimal visible support. There

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3150-575: The mirror image with respect to the vertical center plane corresponds to a shift by one step. Alternating tread stairs are sometimes referred to as "witches stairs", in the supposed belief that they were created during an earlier era as an attempt to repel witches who were thought to be unable to climb such stairs. Such a fanciful origin of the term has since been disproved, with experts finding no mention in any historical literature of stairs that were believed to prevent access by witches. Alternating tread stairs have been in use since at least 1888. Today,

3225-400: The cases r ( φ ) = a φ n {\displaystyle r(\varphi )=a\varphi ^{n}} (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral r = a e k φ {\displaystyle r=ae^{k\varphi }} . The angle α {\displaystyle \alpha } between

3300-410: The choice k = 0.1 , a = 4 , φ ≥ 0 {\displaystyle \;k=0.1,a=4,\;\varphi \geq 0\;} gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius r = a π / 2 {\displaystyle \;r=a\pi /2\;} (diagram, left). For r =

3375-460: The correct foot on the correct step. The slope of alternating tread stairs can be as high as 65° as opposed to standard stairs, which are almost always less than 45°. An advantage of alternating tread stairs is that people can descend while facing forward, in the direction of travel. The only other alternative in such short spaces would be a ladder , which requires a backward-facing descent. Alternating tread stairs may not be safe for small children,

3450-448: The design is used in some loft apartments to access bedrooms or storage spaces. Local building codes often dictate the number of emergency exits required for a building of a given size, including specifying a minimum number of stairwells. For any building bigger than a private house, modern codes invariably specify at least two sets of stairs, completely isolated from each other so that if one becomes impassable due to smoke or flames,

3525-406: The design of the structure. In Scottish architecture, helical stairs are commonly known as a turnpike stair . Helical stairs typically have a handrail along the outer periphery only, and on the inner side may have just a central pole. A "squared helical" stair fills a square stairwell and expands the steps and railing to a square, resulting in unequal steps (wider and longer where they extend into

3600-444: The elderly, or the physically challenged. Building codes typically classify them as ladders, and will only allow them where ladders are allowed, usually basement or attic utility or storage areas infrequently accessed. The block model in the image illustrates the space efficiency gained by an alternating tread stair. The alternating stairs (3) requires one unit of space per step: the same as the half-width stairs (2), and half as much as

3675-424: The elements. Such temporary exposed stairways are sometimes used in conjunction with scaffolding on construction sites, or permanently as an emergency exit . The concept of stairs is believed to be 8000 years old, and are one of the oldest buildings in architectural history. The oldest example of spiral stairs dates back to the 400s BC . Medieval architecture saw experimentation with many different shapes, and

3750-529: The fact that this is often an escape route. This applies to both flammable objects and other objects that may become an obstacle to the rescue work. Many housing cooperatives have general prohibitions against storage in common stairwells. Under stairs there is often a lot of area and volume that is not otherwise used. There are several examples of creative uses, such as wardrobes , home offices , playroom , or general storage such as drawers, cupboards or shelves for storage. Spiral In mathematics ,

3825-473: The first 5 examples: A Cornu spiral has two asymptotic points. The spiral of Theodorus is a polygon. The Fibonacci Spiral consists of a sequence of circle arcs. The involute of a circle looks like an Archimedean, but is not: see Involute#Examples . The following considerations are dealing with spirals, which can be described by a polar equation r = r ( φ ) {\displaystyle r=r(\varphi )} , especially for

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3900-503: The floorplan. These traditional arrangements have the advantage of being easily understood by building occupants and occasional visitors. Some architects save floor footprint space while still meeting the exit requirement, by housing two stairwells in a " double helix " or "scissors stairs" configuration whereby two stairwells occupy the same floor footprint, but are intertwined while being separated by fireproof partitions along their entire run. However, this design deposits anybody descending

3975-403: The full-width stairs (1). Thus, the horizontal distance between steps is in this case reduced by a factor of two, reducing the size of each step. The horizontal distance between steps is reduced by a factor less than two if for construction reasons there are narrow "unused" step extensions. These stairs often (including this example) illustrate the mathematical principle of glide plane symmetry:

4050-484: The groove on one side of a gramophone record closely approximates a plane spiral (and it is by the finite width and depth of the groove, but not by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loops differ in diameter. In another example, the "center lines" of the arms of a spiral galaxy trace logarithmic spirals . The second definition includes two kinds of 3-dimensional relatives of spirals: In

4125-525: The height. Very tall multi-turn helical staircases are usually found in old stone towers within fortifications , churches , and in lighthouses . Winders may be used in combination with straight stairs to turn the direction of the stairs. This allows for a large number of permutations in designs. The earliest known helical staircases appear in Temple A in the Greek colony Selinunte , Sicily , to both sides of

4200-469: The helix needs to be steep to allow the weight to distribute safely down the structure in the most vertical manner possible. Helical steps with center columns or perimeter support do not have this limitation. Building codes may limit the use of helical stairs to small areas or secondary usage, if their treads are not sufficiently wide or have risers taller than 9.5 inches (240 mm). Double helix staircases are possible, with two independent helical stairs in

4275-472: The introduction of kit form helical stairs. Modular, standardized steps and handrails can be bolted together to form a complete unit. These stairs can be made out of steel, timber, concrete, or a combination of materials. Where there is insufficient space for the full run length of normal stairs, "alternating tread stairs" may be used (other names are "paddle stairs", "zig-zag stairs", or "double-riser stairs"). Alternating tread stairs can be designed to allow for

4350-432: The newels. At corners, there are quarter-turn caps . For post-to-post systems, the newels project above the handrails. Another, more classical, form of handrailing that is still in use is the tangent method. A variant of the cylindric method of layout, it allows for continuous climbing and twisting rails and easings. It was defined from principles set down by architect Peter Nicholson in the 18th century. The dimensions of

4425-469: The origin (like a hyperbolic spiral) and approaches the circle with radius r = a π {\displaystyle \;r=a\pi \;} (diagram, right). Two well-known spiral space curves are conical spirals and spherical spirals , defined below. Another instance of space spirals is the toroidal spiral . A spiral wound around a helix, also known as double-twisted helix , represents objects such as coiled coil filaments . If in

4500-469: The other remains usable. The traditional way to satisfy this requirement was to construct two separate stairwell stacks, each occupying its own footprint within each floorplate. Each stairwell is internally configured into an arrangement often called a "U-return" or "return" design. The two stairwells may be constructed next to each other, separated by a fireproof partition, or optionally the two stairwells may be located at some distance from each other within

4575-410: The pattern of florets in the head of a sunflower was proposed by H. Vogel. This has the form where n is the index number of the floret and c is a constant scaling factor, and is a form of Fermat's spiral . The angle 137.5° is the golden angle which is related to the golden ratio and gives a close packing of florets. Spirals in plants and animals are frequently described as whorls . This

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4650-632: The prehistoric Newgrange monument in County Meath , Ireland . Newgrange was built around 3200 BCE, predating the Celts; triple spirals were carved at least 2,500 years before the Celts reached Ireland, but have long since become part of Celtic culture. The triskelion symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures: examples include Mycenaean vessels, coinage from Lycia , staters of Pamphylia (at Aspendos , 370–333 BC) and Pisidia , as well as

4725-415: The same floor footprint. This is often in compliance with legal safety requirements to have two independent fire escape paths. Helical stairs can be characterized by the number of turns that are made. A "quarter-turn" stair deposits the person facing 90° from the starting orientation. Likewise, there are half-turn, three-quarters-turn and full-turn stairs. A continuous helix may make many turns depending on

4800-695: The same vertical space, allowing one person to ascend and another to descend without ever meeting, if they choose different helices. For examples, the Pozzo di San Patrizio allows one-way traffic so that laden and unladen mules can ascend and descend without obstruction, while Château de Chambord , Château de Blois , and the Crédit Lyonnais headquarters ensure separation for social purposes. Emergency exit stairways, though built with landings and straight runs of stairs, are often functionally double helices, with two separate stairs intertwined and occupying

4875-453: The side picture, the black curve at the bottom is an Archimedean spiral , while the green curve is a helix. The curve shown in red is a conical spiral. A two-dimensional , or plane, spiral may be easily described using polar coordinates , where the radius r {\displaystyle r} is a monotonic continuous function of angle φ {\displaystyle \varphi } : The circle would be regarded as

4950-407: The spiral tangent and the corresponding polar circle (see diagram) is called angle of the polar slope and tan ⁡ α {\displaystyle \tan \alpha } the polar slope . From vector calculus in polar coordinates one gets the formula Hence the slope of the spiral r = a φ n {\displaystyle \;r=a\varphi ^{n}\;}

5025-468: The stack into alternating locations on each successive floor, and this can be very disorienting. Some building codes recommend using a color-coded stripe and signage to distinguish otherwise identical-looking stairwells from each other, and to make following a quick exit path easier. Ergonomically and for safety reasons, stairs must have certain dimensions so that people can comfortably use them. Building codes typically specify certain clearances so that

5100-503: The stairs are not too steep or narrow. Staircase A stairwell or stair room is a room in a building where a stair is located, and is used to connect walkways between floors so that one can move in height. Collectively, a set of stairs and a stairwell is referred to as a staircase or stairway . In buildings with several housing units, a stairway can be a necessary common area for getting to and from apartments . Staircases provide vertical access to connected floors in

5175-424: The stairwell should then be a separate fire compartment separated from the rest of the building via fire doors . For fire safety it can be advantageous that both the walls, ceiling and the stairs themselves in a stairwell are made of non-combustible materials, and be illuminated with daylight during the day and artificial lighting at darkness. Storage of objects in shared stairwells is generally prohibited due to

5250-506: The standard spirals r ( φ ) {\displaystyle r(\varphi )} is either a power function or an exponential function. If one chooses for r ( φ ) {\displaystyle r(\varphi )} a bounded function, the spiral is bounded, too. A suitable bounded function is the arctan function: Setting r = a arctan ⁡ ( k φ ) {\displaystyle \;r=a\arctan(k\varphi )\;} and

5325-405: The terms "helix" and "helical" to describe circular stairways more clearly and precisely, while reserving the term "spiral" for a curve restricted to a flat plane. Helical stairs, sometimes referred to in architectural descriptions as vice , wind around a newel (also called the "central pole"). The presence or absence of a central pole or newel does not affect the overall terminology applied to

5400-415: The top and bottom. However, many modern architects may not choose straight flights of stairs because: Another form of a straight staircase is the "space saver staircase", also known as "paddle stairs" or "alternating tread staircases". These designs can be used for a steeper rise, but they can only be used in certain circumstances, and must comply with regulations. However, a basic straight flight of stairs

5475-426: The twisting curvilinear shape as an embellishment, either within or outside of their buildings. Helical stairs have the disadvantage of being very steep if they are tight (small radius) or are otherwise not supported by a center column. The cylindrical spaces they occupy can have a narrow or wide diameter: "Open well" helical or circular stairs designed by architects often do not have a central pole, but there usually

5550-441: The unit circle has in polar coordinates the simple description:   ( r , φ ) ↦ ( 1 r , φ )   {\displaystyle \ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ } . The function r ( φ ) {\displaystyle r(\varphi )} of a spiral is usually strictly monotonic, continuous and un bounded . For

5625-476: The upper or lower floors of a building, respectively. Each step is composed of a tread and a riser . Some treads may include a nosing . The balustrade is the system of railings and balusters that prevents people from falling over the edge. Handrails may be continuous (sometimes called over-the-post ) or post-to-post (or more accurately newel-to-newel ). For continuous handrails on long balconies, there may be multiple newels and tandem caps to cover

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