"ordinary cube has 8 corners, they could be labeled A to H, with A opposite H, B opposite G, and so on. In a hemicube, A and H would be treated as the same corner. So would B and G, and so on. The edge AB would become the same edge as GH, and the face ABEF would become the same face as CDGH. The new shape has only three faces, 6 edges and 4 corners. The 11-cell cannot be formed with regular geometry in flat (Euclidean) hyperspace, but only in positively curved (elliptic) hyperspace. A few years after Grünbaum's"
"Clock face A clock face, or dial, is the part of an analog clock (or watch) that displays the time through the use of a fixed-numbered dial or dials and moving hands. In its most basic form, recognized throughout the world, the periphery of the dial is numbered 1 through 12 indicating the hours in a 12-hour cycle, and a short ""hour hand"" makes two revolutions in a day. A long ""minute hand"" makes one revolution every hour. The face may also include a ""second hand"", which makes one revolution per minute. The term is less commonly used for the"
"ring formed by a 12-hour clock is typically denoted formula_1 or, less often, formula_2. In this case, 11+3=2. (The same idea can also be expressed using overbars and subscripts, as formula_3). However, in the case of 24-hour clock, formula_4 and 11+3=14 (i.e., formula_5). Generalising the previous case, when is a set (mathematics) and an equivalence relation on it, denotes the quotient set, which is the set of the equivalence classes of under ; these are the subsets of consisting of all elements related by to (any) one of their members. The slash, sometimes distinguished as ""forward slash"", is used in"
"have a ""second hand"" which indicates the seconds in the current minute. The only other widely used clock face today is the 24 hour analog dial, because of the use of 24 hour time in military organizations and timetables. Before the modern clock face was standardized during the Industrial Revolution, many other face designs were used throughout the years, including dials divided into 6, 8, 10, and 24 hours. During the French Revolution the French government tried to introduce a 10-hour clock, as part of their decimal-based metric system of measurement, but it didn't catch on. An Italian 6 hour"
"of order 2""n"", except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has three versions of D as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D as subgroups. Uniform star antiprisms are named by their star polygon bases, {""p""/""q""}, and exist in prograde and retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by inverted fractions, ""p""/(""p"" - ""q"") instead of ""p""/""q"", e.g. 5/3 instead of 5/2. In the retrograde forms but not in the prograde"
"has 27 vertices, 72 3-edges, and 27 3{3}3 faces. Its complex reflection group is [3][3], order 648. The 2 is fourth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes. The 2 polytope is fourth in dimensional series 2. The 2 polytope is second in dimensional series 2. The rectified 2 has 216 vertices, and 126 facets: 72 rectified 5-simplices, and 27 rectified 5-orthoplexes and 27 5-demicubes . Its vertex figure is"
"In 1915, the complex 132 strike mechanism was disabled, due to the wartime curfew. Today the clock displays the original I to XXIIII numbering around the outside, with I at the right hand side. The gilded stars are purely decorative. The signs of the zodiac are in anticlockwise order around the inner zodiac dial: the zodiac wheel rotates clockwise with the hour hand but very slightly faster. As a result, the hour hand moves slowly anticlockwise relative to the zodiac, so that it passes through each sign in the course of the year. In 1996, another restoration was initiated, to"
"typefaces for the date, but most of the digits have been old-style numerals, except with the 4 and 8 neither ascending nor descending. The only significant divergence is that the small 3 was non-descending (the same size as a 0, 1, or 2) in the early history, before switching to a descending, large 3 for just one year in 1934 and then permanently () in 1943. Similarly, the digit 5 was small and non-descending up to 1945, then from 1950 and on, it became a large descending 5. From 1959 until 2008, the Lincoln Memorial was shown on the reverse"
"time display on digital clocks and watches. A second type of clock face is the 24-hour analog dial, widely used in military and other organizations that use 24-hour time. This is similar to the 12-hour dial above, except it has hours numbered 1–24 around the outside, and the hour hand makes only one revolution per day. Some special-purpose clocks, such as timers and sporting event clocks, are designed for measuring periods less than one hour. Clocks can indicate the hour with Roman numerals or Hindu–Arabic numerals, or with non-numeric indicator marks. The two numbering systems have also been used in"
"balancing dials give local sidereal time in degrees, minutes, and seconds of arc. An aperture indicates the name of the day of the week. The rear planisphere, similar in size to the one in the front but strictly circular, displays the brightest stars of the main circumpolar constellations (Ursa Major, Ursa Minor, and Cassiopeia). It is centred on the north celestial pole and, with its encircling silvered ring of 24-hour intervals, rotates once a sidereal day relative to a fixed meridian-index. Damage to the clock's mechanism was sustained during the fire of 9 July 1984; after 10 years' reparation work,"
"combination, with the prior indicating the hour and the latter the minute. Longcase clocks (grandfather clocks) typically use Roman numerals for the hours. Clocks using only Arabic numerals first began to appear in the mid-18th century. The clock face is so familiar that the numbers are often omitted and replaced with applied indices (undifferentiated hour marks), particularly in the case of watches. Occasionally, markings of any sort are dispensed with, and the time is read by the angles of the hands. Most modern clocks have the numbers 1 through 12 printed at equally spaced intervals around the periphery of the"
") follows a letter, it denotes an anticlockwise face turn; while a letter without a prime symbol denotes a clockwise turn. These directions are as one is looking at the specified face. A letter followed by a 2 (occasionally a superscript ) denotes two turns, or a 180-degree turn. ""R"" is right side clockwise, but ""R′"" is right side anticlockwise. The letters ""x"", ""y"", and ""z"" are used to indicate that the entire Cube should be turned about one of its axes, corresponding to R, U, and F turns respectively. When ""x"", ""y"" or ""z"" are primed, it is an"
"right but directed down, is clockwise. Such non-standard orientations are rarely used in mathematics but are common in 2D computer graphics, which often have the origin in the top left corner and the down the screen or page. See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix. Particularly useful are the matrices formula_13, formula_14, formula_15 for 90°, 180°, and 270° counter-clockwise rotations. Since formula_16 the plane of matrices formula_17 is isomorphic to the complex number plane ℂ, and the above rotation matrix is a point on its unit circle, which"
"hand because it goes round 60 times faster than the minute hand and 720 times faster than the hour hand. Some Analog watches don't have numbers printed onto them. The numbers can be determined because the strap is always at 12 & 6. Often only the 12 o'clock position is marked, or only the 12, 3, 6 & 9 positions. An analog watch can be used to locate north and south. The Sun appears to move in the sky over a 24-hour period while the hour hand of a 12-hour clock takes twelve hours to complete one rotation. In the"
"numbers from 0 to 23 (or 1 to 24) to mark the day is the ""24-hour clock system"". Sundials use 24-hour analog dials—the shadow traces a path that repeats approximately once per day. Many sundials are marked with the double-XII or double-12 system, in which the numbers I to XII (or 1 to 12) are used twice, once for the morning hours, and once for the afternoon and evening hours. So VI (or 6) appears twice on many dials, once near sunrise and once near sunset. Modern 24-hour analog dials—other than sundials—are almost always marked with 24 numbers or hour"
"example, the snub cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration. The notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol {5/2}, meaning it has 5 sides going around the centre twice. For example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of {5/2,5} which expands to an explicit vertex configuration 5/2.5/2.5/2.5/2.5/2 or combined as (5/2). The great stellated dodecahedron, {5/2,3} has a triangular vertex figure and configuration"
"appears below before the 2 starts rotating slowly. The blue 2 rotates to reveal a red 2, green 2 and a white 2. The ident would continually spin after this. In 1969, when BBC1 launched its own colour service, the ident was updated to match BBC1's presentation. The 2 was now all white on a blue background, with the BBC2 logo underneath the 2. The clock that went alongside this look was originally a traditional clock face with Roman numerals before it was changed to a light blue clockface on a dark blue background with BBC2 legend below. This clock"
"F, B, U, and D indicate a clockwise quarter turn of the left, right, front, back, up, and down face respectively. Half turns are indicated by appending a 2. A counterclockwise quarter turn is indicated by appending a prime symbol ( ′ ). The letters M, S and E are used to denote the turning of a middle layer. M represents turning the layer between the R and L faces 1 quarter turn top to bottom. S represents turning the layer between the F and B faces 1 quarter turn clockwise, as seen from the front. E represents turning the"
"the calendar hand has to be reset manually each leap year on 29 February. A moon dial circles the inner ring of the zodiac, displaying the moon phase. The principal hand of the clock indicates the time of day on the outer ring of 24 golden Roman numerals, which run twice from I to XII. It features two suns, the smaller one indicating the date on the ""rete""'s calendar dial. The larger one circles the zodiac at one revolution per year and also rotates across the planisphere once per day. Its crossing of the horizon and dawn lines twice per"
"counterclockwise, with the 0 being at about 5 o'clock. A curved device called a finger stop sits above the dial at approximately the 4 o'clock position. The physical nature of the dialing mechanism on rotary phones allowed the use of physical locking mechanisms to prevent unauthorized use. The lock could be integral to the phone itself or a separate device inserted through the finger hole nearest the finger stop to prevent the dial from rotating. In addition to the numbers, the faceplate is often printed with letters corresponding to each finger hole. In North America, traditional dials have letter codes"
"One solution of this variation allows displaying the first three letters of any month and relies on the fact that lower-case letters ""u"" and ""n"" and also ""p"" and ""d"" are inverses of each other. A number of variations are manufactured and sold as souvenirs, differing in the appearance and the existence of additional bars to set the current month. Two-cube calendar A two-cube calendar is a desk calendar consisting of two cubes with faces marked by digits 0 through 9. Each face of each cube is marked with a single digit, and it is possible to arrange the cubes"
"pocket watch. An open-faced, or Lépine, watch, is one in which the case lacks a metal cover to protect the crystal. It is typical for an open-faced watch to have the pendant located at 12:00 and the sub-second dial located at 6:00. Occasionally, a watch movement intended for a hunting case (with the winding stem at 3:00 and sub second dial at 6:00) will have an open-faced case. Such watch is known as a ""sidewinder."" Alternatively, such a watch movement may be fitted with a so-called conversion dial, which relocates the winding stem to 12:00 and the sub-second dial to"
"dial, resulting in the name ""California dial"". A modern example of a watch with a California dial is Nomos Glash%C3%BCtte's ""Club Campus"". California dial In watch-making, or horology, a California dial refers to a clock face that consists of half Roman (usually 10 o'clock to 2 o'clock) and half Arabic numerals (usually 4 o'clock to 8 o'clock). Sometimes the hours of 3, 6, and 9 are replaced with a dash instead of a numeral and an inverted triangle in place of the 12. The use of this dial style dates back to the 1930s, for example in early Rolexes, Panerai,"
"are original and date from the 1490s), which revolve slightly more slowly than the pointer to show the position of the sun in the zodiac. In the middle of the clockface is the earth (in the centre) and the moon, which revolves to show its phases, surrounded by stars which are fixed in position. The background is of blue enamel. The smaller blue circles in the four corners are not now used. Below the clock is the archway, two storeys high, through which the street known as the Merceria leaves the Piazza on its way to the Rialto (This section"
"the shadow is observed from the other side of the dial's plane and is observed as moving in the opposite direction. Some clocks were constructed to mimic this. The best-known surviving example is the astronomical clock in the Münster Cathedral, whose hands move counterclockwise. Occasionally, clocks whose hands revolve counterclockwise are nowadays sold as a novelty. Historically, some Jewish clocks were built that way, for example in some synagogue towers in Europe such as the Jewish Town Hall in Prague, to accord with right-to-left reading in the Hebrew language. In 2014 under Bolivian president Evo Morales, the clock outside the"
"Octagonal antiprism In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals. In the case of a regular 8-sided base, one usually considers the case where its copy is twisted by an angle 180°/""n"". Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it"
"for clockwise, – for anticlockwise and 2 for 180-degree turns. Another notation appeared in the 1982 ""The Ideal Solution"" book for Rubik's Revenge. Horizontal planes were noted as tables, with table 1 or T1 starting at the top. Vertical front to back planes were noted as book, with book 1 or B1 starting from the left. Vertical left to right planes were noted as windows, with window 1 or W1 starting at the front. Using the front face as a reference view, table moves were left or right, book moves were up or down, and window moves were clockwise or"
"human readable form. Clocks can be classified by the type of time display, as well as by the method of timekeeping. Analog clocks usually use a clock face which indicates time using rotating pointers called ""hands"" on a fixed numbered dial or dials. The standard clock face, known universally throughout the world, has a short ""hour hand"" which indicates the hour on a circular dial of 12 hours, making two revolutions per day, and a longer ""minute hand"" which indicates the minutes in the current hour on the same dial, which is also divided into 60 minutes. It may also"
"lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G. On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top"
"Decagonal antiprism In geometry, the decagonal antiprism is the eighth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals. In the case of a regular 10-sided base, one usually considers the case where its copy is twisted by an angle 180°/""n"". Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it"
"= √ 2 centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes. The edge vectors have Cartesian coordinates and permutations of these. The face normals (normalized cross products of edges that share a common vertex) of the 6 square faces are , and . The face normals of the 8 hexagonal faces are . The dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either − or −. The dihedral angle is approximately 1.910633 radians (109.471° )"
"subtitled programme would be accompanied by the additional caption below the numeral stating 'Ceefax 270' and later following a change of page 'Ceefax 888'. The clock which accompanied this new look was located in the BBC Two Noddy room and comprised an orange clock face with counters of ever increasing thickness and with a 'polo' mint centre upon a black background. A plain orange and white 2D striped '2' logo was placed below the clock. The use of this old clock was due to the fact that the technology did not exist for an electronic timepiece, and would not until"
"horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal, a 6 that looks like a ""b"" is not practical. Just as in most modern typefaces, in typefaces with text figures the 6 character usually has an ascender, as, for example, in . This numeral resembles an inverted ""9"". To disambiguate the two on objects and documents that can be inverted, the 6 has often been underlined, both in handwriting and on printed labels. 6 6 (six) is the natural number following 5 and preceding 7. The SI prefix for 1000 is exa-"