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30-507: The first building was constructed by Jakob Friedrich von Batzendorf. The city was planned with the tower of the palace ( Schloss ) at the centre and 32 streets radiating out from it like spokes on a wheel , or ribs on a folding fan , so that a nickname for Karlsruhe in German is the "fan city" ( Fächerstadt ). Originally partially made of wood, the palace had to be rebuilt in 1746, using stone. Charles Frederick, Margrave of Baden-Durlach at

60-497: A of 67.5°. Regarding r 3 : The size of the spoke holes in the flange does not matter for the needed spoke length. This term removes the effect of the hole size. Since the holes are usually small (just over 2 mm in diameter), the effect is small and in practice matters little. For radially spoked wheels (zero crossings), the formula simplifies to the Pythagorean theorem , with spoke length l plus r 3 being

90-437: A "Z" to keep it from pulling through its hole in the hub. The bent version has the advantage of replacing a broken spoke in a rear bicycle wheel without having to remove the rear gears . Wire wheels, with their excellent weight-to-strength ratio, soon became popular for light vehicles. For everyday cars, wire wheels were soon replaced by the less expensive metal disc wheel, but wire wheels remained popular for sports cars up to

120-511: A good quality spoke is capable of supporting about 225 kgf (c. 500 pounds-force or 2,200 newtons ) of tension, they are used at a fraction of this load to avoid suffering fatigue failures. Since bicycle and wheelchair wheel spokes are only in tension, flexible and strong materials such as synthetic fibers, are also occasionally used. Metal spokes can also be ovalized or bladed to reduce aerodynamic drag, and butted (double or even triple) to reduce weight while maintaining strength. A variation on

150-409: A regular polygon is also called apothem . In graph theory , the radius of a graph is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph. The radius of the circle with perimeter ( circumference ) C is For many geometric figures, the radius has a well-defined relationship with other measures of the figure. The radius of a circle with area A

180-412: A regular polygon with n sides of length s is given by r = R n s , where R n = 1 / ( 2 sin ⁡ π n ) . {\displaystyle R_{n}=1\left/\left(2\sin {\frac {\pi }{n}}\right)\right..} Values of R n for small values of n are given in the table. If s = 1 then these values are also

210-453: A width of r 1 sin( a ). Equivalently, the law of cosines may be used to first compute the length of the spoke as projected on the wheel's plane (as illustrated in the diagram), followed by an application of the Pythagorean theorem . Radius In classical geometry , a radius ( pl. : radii or radiuses ) of a circle or sphere is any of the line segments from its center to its perimeter , and in more modern usage, it

240-401: Is The radius of the circle that passes through the three non- collinear points P 1 , P 2 , and P 3 is given by where θ is the angle ∠ P 1 P 2 P 3 . This formula uses the law of sines . If the three points are given by their coordinates ( x 1 , y 1 ) , ( x 2 , y 2 ) , and ( x 3 , y 3 ) , the radius can be expressed as The radius r of

270-489: Is a tool originally developed for this purpose. Eventually, the term spoke was more commonly applied to the finished product of the wheelwright 's work than to the materials they used. The spoked wheel was invented to allow the construction of lighter and swifter vehicles. The earliest physical evidence for spoked wheels were found in the Sintashta culture , dating to c. 2000 BCE. Soon after this, horse cultures of

300-484: Is also their length. The name comes from the Latin radius , meaning ray but also the spoke of a chariot wheel. The typical abbreviation and mathematical variable symbol for radius is R or r . By extension, the diameter D is defined as twice the radius: If an object does not have a center, the term may refer to its circumradius , the radius of its circumscribed circle or circumscribed sphere . In either case,

330-398: Is one of some number of rods radiating from the center of a wheel (the hub where the axle connects), connecting the hub with the round traction surface. The term originally referred to portions of a log that had been riven (split lengthwise) into four or six sections. The radial members of a wagon wheel were made by carving a spoke (from a log) into their finished shape. A spokeshave

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360-440: Is sometimes referred to as the angular position or as the azimuth . The radius and the azimuth are together called the polar coordinates , as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane. The third coordinate may be called the height or altitude (if the reference plane is considered horizontal), longitudinal position , or axial position . In

390-403: Is the intersection between the reference plane and the axis. The axis is variously called the cylindrical or longitudinal axis, to differentiate it from the polar axis , which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. The distance from the axis may be called the radial distance or radius , while the angular coordinate

420-636: The Caucasus region used horse-drawn spoked-wheel war chariots for the greater part of three centuries. They moved deep into the Greek peninsula , where they joined with the existing Mediterranean peoples to give rise, eventually, to classical Greece after the breaking of Minoan dominance and consolidations led by pre-classical Sparta and Athens . Neo- Chalcolithic /proto-historic period (1800-1200 BCE) paintings in various regions of India such as Chibbar Nulla, Chhatur Bhoj Nath Nulla, Kathotia, etc. depict

450-441: The ray from the pole in the fixed direction is the polar axis . The distance from the pole is called the radial coordinate or radius , and the angle is the angular coordinate , polar angle , or azimuth . In the cylindrical coordinate system, there is a chosen reference axis and a chosen reference plane perpendicular to that axis. The origin of the system is the point where all three coordinates can be given as zero. This

480-420: The 1960s. Spoked wheels are still popular on motorcycles and bicycles . When building a bicycle wheel , the spokes must have the correct length, otherwise there may not be enough threads engaged, producing a weaker wheel, or they may protrude through the rim and possibly puncture the inner tube. For bicycle spokes, the spoke length is defined from the flange seat to the thread tip. For spokes with bent ends,

510-415: The hub is rotated with reference to the rim one "angle between adjacent flange holes". Thus, multiplying the "angle between adjacent flange holes" by k gives the angle a . For example, a 32 spoke wheel has 16 spokes per side, 360° divided by 16 equals 22.5°. Multiply 22.5° ("angle between adjacent flange holes") by the number of crossings to get the angle a —if 3-crosses, the 32 spoke wheel has an angle

540-541: The last monarch, Frederick II, Grand Duke of Baden , had to move out. The former residence of the rulers of Baden is now used as the Badisches Landesmuseum Karlsruhe. Much of the city centre, including the palace, was reduced to rubble by Allied bombing during World War II but was quickly rebuilt after the war. 49°00′49″N 8°24′16″E  /  49.0136°N 8.4044°E  / 49.0136; 8.4044 Spokes A spoke

570-409: The nominal spoke length does not include the width of the spoke at the bent end. For wheels with crossed spokes (which are the norm), the desired spoke length is where Regarding d : For a symmetric wheel such as a front wheel with no disc brake, this is half the distance between the flanges. For an asymmetric wheel such as a front wheel with disc brake or a rear wheel with chain derailleur ,

600-748: The outside of the vehicle, to prevent wobbling. Also, the dishing allows the wheel to compensate for expansion of the spokes due to absorbed moisture by dishing more. For use in bicycles , heavy wooden-spoked wheels were replaced by lighter wheels with spokes made of tensioned, adjustable metal wires, called wire wheels . These are also used in wheelchairs , motorcycles , automobiles , and early aircraft . Some types of wheels have removable spokes that can be replaced individually if they break or bend. These include bicycle and wheelchair wheels. High quality bicycles with conventional wheels use spokes of stainless steel , while cheaper bicycles may use galvanized (also called "rustless") or chrome plated spokes. While

630-409: The radii of the corresponding regular polygons. The radius of a d -dimensional hypercube with side s is The polar coordinate system is a two - dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The fixed point (analogous to the origin of a Cartesian system ) is called the pole , and

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660-414: The radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The inradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow object is the radius of its cavity. For regular polygons , the radius is the same as its circumradius. The inradius of

690-414: The rim to the brake in the opposite direction—(via the hub) when braking. Constructing a tension-spoked wheel from its constituent parts is called wheelbuilding and requires the correct building procedure for a strong and long-lasting end product. Tensioned spokes are usually attached to the rim or sometimes the hub with a spoke nipple . The other end is commonly peened into a disk or uncommonly bent into

720-440: The slope, r 2 minus r 1 being the base, and d being the rise: The spoke length formula computes the length of the space diagonal of an imaginary rectangular box . Imagine holding a wheel in front of you such that a nipple is at the top. Look at the wheel from along the axis. The spoke through the top hole is now a diagonal of the imaginary box. The box has a depth of d , a height of r 2 - r 1 cos( α ) and

750-462: The time and who eventually became Charles Frederick, Grand Duke of Baden , then had the palace altered by Balthasar Neumann and Friedrich von Kesslau until 1770, adding larger windows and doors, pavilions and wings. In 1785, Wilhelm Jeremias Müller shortened the tower, adding a cupola. During the Revolutions of 1848 , Leopold, Grand Duke of Baden , was expelled in 1849 for some time. In 1918,

780-445: The usage of chariots with spoked wheels. Celtic chariots introduced an iron rim around the wheel in the 1st millennium BCE. The spoked wheel was in continued use without major modification until the 1870s, when wire wheels and rubber tires were invented. Spokes can be made of wood, metal, or synthetic fiber depending on whether they will be in tension or compression . The original type of spoked wheel with wooden spokes

810-441: The value of d is different for the left and right sides. a is the angle between (1) the radius to a nipple hole in the rim to which a spoke is attached, and, (2) the radius to the flange hole holding the spoke. The spoke crosses either 1, 2, or 3 oppositely pointing spokes depending on the lacing design. On the flange, the angle between the radii of adjacent holes is 360°/ m (for equally spaced holes). For each spoke crossed,

840-418: The wire-spoked wheel was Tioga's "Tension Disk", which appeared superficially to be a solid disk but was in fact constructed using the same principles as a normal tension-spoked wheel. Instead of individual wire spokes, a continuous thread of Kevlar ( aramid ) was used to lace the hub to the rim under high tension. The threads were encased in a translucent disk for protection and some aerodynamic benefit, but this

870-404: Was not a structural component. Wire spokes can be radial to the hub but are more often mounted tangentially to the hub. Tangential spoking allows for the transfer of torque between the rim and the hub. Tangential spokes are thus necessary for the drive wheel, which has torque at the hub from pedalling, and any wheels using hub-mounted brakes such as disk or band brakes, which transfer torque from

900-424: Was used for horse -drawn carriages and wagons . In early motor cars, wooden spoked wheels of the artillery type were normally used. In a simple wooden wheel, a load on the hub causes the wheel rim to flatten slightly against the ground as the lowermost wooden spoke shortens and compresses. The other wooden spokes show no significant change. Wooden spokes are mounted radially . They are also dished, usually to

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