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Devil's staircase

The Devil's staircase, also known as the Cantor function or Cantor ternary function, is a continuous and monotonically increasing function f: [0,1] \to [0,1] that maps the unit interval onto itself, starting at f(0) = 0 and ending at f(1) = 1, while remaining constant on each interval in the complement of the ternary Cantor set and thus having derivative zero almost everywhere.^[1]^[2] Its graph resembles an infinite staircase with flat steps corresponding to the removed intervals in the Cantor set construction, earning it the evocative name due to this "diabolical" structure that defies intuitive notions of monotonic increase.^[2]^[3] Introduced by Georg Cantor in his 1883 paper on the Cantor set, the function serves as a foundational example of a singular function—one that is continuous and of bounded variation but not absolutely continuous, highlighting pathologies in real analysis.^[4] To construct it explicitly, express x \in [0,1] in its ternary (base-3) expansion; if x has a 1 in its expansion (placing it outside the Cantor set), replace the first 1 with a 0 and all subsequent 2s with 1s, then interpret the result as a binary (base-2) number to obtain f(x); for points in the Cantor set (with expansions using only 0s and 2s), map 2s to 1s and read as binary.^[1]^[2] This yields a self-similar function satisfying functional equations like f(x/3) = f(x)/2 and f((2 + x)/3) = 1/2 + f(x)/2, with symmetry f(1 - x) = 1 - f(x).^[1] Key properties include its uniformity of continuity despite the staircase appearance, the fact that it maps the Cantor set (of Lebesgue measure zero) onto [0,1], and its role as a counterexample showing that monotonic functions need not be absolutely continuous or differentiable almost everywhere.^[2]^[3] The function is strictly increasing overall but flat on a set of positive measure (the complement of the Cantor set, with total measure 1), making it a prototype for studying measure-preserving transformations and fractal dimensions.^[1] It also appears in probability as a distribution function with no atoms yet singular with respect to Lebesgue measure, and its moments can be computed via integrals involving binomial coefficients and Bernoulli numbers.^[1]^[5] More broadly, "devil's staircase" denotes a class of similar functions in mathematics, characterized by being non-decreasing, continuous, constant on intervals of total length equal to the domain's measure, and increasing by the domain's length overall, often arising as cumulative distribution functions of singular measures supported on sets of measure zero.^[6] Generalizations include the q-adic versions F_q for bases q > 2, Minkowski's question-mark function linking quadratic irrationals to rationals, and occurrences in dynamical systems where the rotation number \rho as a function of a parameter (e.g., in circle maps or billiards) forms a devil's staircase with plateaus at rational values and fractal structure of dimension approximately 0.87.^[1]^[2] These structures appear in physics, such as in the fractional quantum Hall effect, underscoring their interdisciplinary significance.^[2]

In mathematics

Definition and construction

The devil's staircase, also known as the Cantor function or Cantor–Lebesgue function, is a continuous, non-decreasing function f: [0,1] \to [0,1] that remains constant on each of the open intervals removed during the construction of the Cantor set, while surjectively mapping the Cantor set itself onto the entire interval [0,1].^[3] This function was introduced by Georg Cantor in a 1883 paper, where it served as a counterexample to an extension of the fundamental theorem of calculus to discontinuous functions, as noted by Scheeffer.^[7]^[3] The construction proceeds iteratively alongside that of the Cantor set, beginning with the unit interval C_0 = [0,1]. At the first stage, remove the open middle-third interval (1/3, 2/3) to obtain C_1 = [0,1/3] \cup [2/3,1], and define the function to take the constant value f(x) = 1/2 on the removed interval. In the next stage, remove the open middle-third intervals from each remaining segment, yielding C_2 consisting of four closed intervals of length $1/9, and extend the function by setting it to $1/4 on (1/9, 2/9), $3/4 on (7/9, 8/9), while the previously removed interval remains at $1/2. This process continues infinitely: at stage n, remove the middle-third open intervals from each of the $2^{n-1} segments of C_{n-1} to form C_n, and assign constant values to these removed intervals such that the function increases only on the Cantor set C = \bigcap_{n=0}^\infty C_n, which has Lebesgue measure zero. The resulting function f is the pointwise limit of these partial functions and is continuous and non-decreasing with f(0) = 0 and f(1) = 1.^[2]^[8] An explicit formula for the devil's staircase can be given using ternary expansions. For x \in [0,1], write x = \sum_{n=1}^\infty a_n / 3^n where each a_n \in \{0,1,2\}, choosing the expansion that does not end in infinite 2's when possible. Then define f(x) = \sum_{n=1}^\infty b_n / 2^n, where the sequence (b_n) is obtained from (a_n) by replacing 0s with 0s, 1s with 0s, and 2s with 1s, and interpreting the result as a binary expansion. For points in the Cantor set (those with no 1's in their ternary expansion), this simply reinterprets the ternary digits (with 2's as 1's) as a binary number.^[3]^[8] Visually, the graph of f resembles a monotonic increasing staircase, with flat horizontal steps corresponding to the constant values on the removed intervals (whose total length sums to 1) and the entire vertical rise of 1 accumulated continuously across the uncountably many points of the Cantor set.^[2]

Properties and applications

The devil's staircase, also known as the Cantor function, is a continuous function f: [0,1] \to [0,1] that is non-decreasing, mapping the unit interval onto itself while remaining constant on each interval in the complement of the ternary Cantor set, which has Lebesgue measure 1.^[9] This constancy implies that the derivative f'(x) = 0 almost everywhere on [0,1], yet the function satisfies f(1) - f(0) = 1, exemplifying a pathological case where the total increase occurs despite vanishing derivative on a set of full measure.^[9] As a monotone function, it has bounded variation with total variation equal to 1, but this variation is concentrated on the Cantor set of measure zero.^[9] A hallmark of the devil's staircase is its singularity: it is a singular continuous function, meaning it is continuous and non-constant but not absolutely continuous.^[9] Specifically, it maps the Cantor set—a compact, nowhere dense set of Lebesgue measure zero—onto the entire interval [0,1] of positive measure, thereby distorting Lebesgue measure and failing the absolute continuity condition that requires the image of measure-zero sets to have measure zero.^[9] This property underscores its role as a counterexample in real analysis, challenging intuitions about monotonic functions; for instance, it shows that continuity and monotonicity do not imply absolute continuity, and it violates expectations in Harnack's theorem regarding the integral representation of such functions.^[9] In measure theory and the Lebesgue integral, the devil's staircase illustrates key limitations: it cannot be expressed as the indefinite integral of a Lebesgue integrable function \phi, since \int_0^1 |f'(x)| \, dx = 0 while the total increase is 1, highlighting the distinction between bounded variation and absolute continuity.^[9] Furthermore, its associated singular measure (the Stieltjes measure induced by f) is used in studying Fourier-Stieltjes transforms, where the function's self-similarity and support on the Cantor set provide insights into the spectral properties of singular distributions orthogonal to Lebesgue measure.^[9] These features make it a foundational example for exploring pathological behaviors in integration theory and harmonic analysis.^[9]

Places

In the United Kingdom

In the Scottish Highlands, particularly around Glencoe, the term "Devil's Staircase" refers to two challenging geographical features renowned for their steep ascents and rugged terrain, attracting hikers and mountaineers seeking dramatic Highland scenery. These paths, both evoking the imagery of infernal climbs due to their relentless gradients and zigzag routes, form integral parts of iconic walking routes in the region.^[10] One prominent Devil's Staircase is the approximately 2-mile section of the West Highland Way, a long-distance trail stretching from Milngavie to Fort William. This path involves a steep ascent and descent between Glencoe (starting near Altnafeadh) and Kinlochleven, featuring an elevation gain of around 300 meters to reach the pass at 550 meters above sea level. Originally constructed in 1752 as part of a military road network by Major William Caulfeild to facilitate troop movements and control in the Highlands following Jacobite unrest, the route follows a historic zigzag stone track built by soldiers. The path offers panoramic views of the surrounding Munros, including Buachaille Etive Mòr and the Aonach Eagach ridge, and is traversed by approximately 50,000 hikers annually as part of the broader West Highland Way experience.^[11]^[12]^[13] The other notable feature is the Devil's Staircase path on the Aonach Eagach ridge, a steep and rocky scramble in the Glencoe area that serves as the eastern access to one of Scotland's most demanding ridge traverses. This 5-6 mile route along the notched quartzite ridge, from the col at the Devil's Staircase eastward to the Pap of Glencoe, includes exposed sections with technical scrambling over pinnacles and narrow edges, demanding sure-footedness and vertigo tolerance. Known for its dramatic exposure and as a classic test of mountaineering skills, the ridge has been popular among hillwalkers since the late 19th century, particularly with the rise of Munro bagging after the publication of Munro's Tables in 1891, and it encompasses two Munros: Meall Dearg and Sgorr nam Fiannaidh.^[14]^[15] Both paths share a reputation for their "hellish" steepness, with zigzag constructions that mimic staircases and test physical endurance, navigation in variable weather, and basic fitness levels, though they remain accessible to prepared walkers without requiring specialized climbing gear. While comparable in challenge to other UK trails like the Cuillin ridges on Skye, these Scottish features stand out for their historical military origins and integration into modern long-distance hiking culture.^[10]^[11]

In New Zealand

The Devil's Staircase is a scenic winding section of State Highway 6 (SH6) in the Otago region, hugging the western shore of Lake Wakatipu between Queenstown and Kingston. Spanning approximately 12 km, it features multiple hairpin turns and gentle gradients with dramatic drops to the lake, offering stunning views of the lake's fjord-like waters, the Remarkables mountain range, and surrounding alpine scenery. This picturesque stretch is part of the Southern Scenic Route, renowned for its accessibility and photogenic pullouts.^[16]^[17] Originating as a rugged bridle path in the 1860s during the Otago gold rush, the route was used by travelers, herders, and miners navigating the steep, narrow terrain along the lake—earning its name from the "diabolical" challenges of the era, including tight switchbacks and exposure to weather. Paved and improved in the early 20th century as part of New Zealand's expanding road network, it now provides a smooth drive, though its curves demand caution, especially for cyclists and larger vehicles. Historical accounts note frequent hazards like slips and isolation in early days, but modern engineering has made it a highlight for tourists.^[18]^[19] As a key link on the route from Queenstown to Te Anau and Fiordland, the Devil's Staircase supports tourism to South Island attractions, with SH6 in the Queenstown area handling over 500,000 vehicles annually as of 2023. Its allure draws road trippers for the panoramic vistas, but advisories from NZ Transport Agency emphasize slow speeds on blind corners, no shoulders in places, and seasonal caution for ice or wind.^[20]^[21]

In the United States

The Devil's Staircase Wilderness spans approximately 30,000 acres in the Siuslaw National Forest within Oregon's Coast Range, encompassing some of the last intact old-growth forests in the region. Designated as wilderness in 2019 under the John D. Dingell, Jr. Conservation, Management, and Recreation Act, the area protects diverse ecosystems featuring towering Douglas-fir, western hemlock, and cedar trees amid rugged canyons and river corridors along the Umpqua and Smith Rivers.^[22] This remote terrain supports vital wildlife, including black bears, Roosevelt elk, and seasonal salmon runs in the rivers, while emphasizing conservation of pristine habitats free from motorized access or development. Lacking any maintained trails to preserve its wild character, the wilderness is accessible primarily by boat along the rivers or through off-trail bushwhacking, offering solitude for experienced adventurers seeking untrammeled natural immersion.^[22] In Janesville, Wisconsin, the Devil's Staircase forms a 1.5-mile segment of the Ice Age National Scenic Trail within Riverside Park, providing an easy, family-oriented riverside path along the Rock River. Established as part of the broader trail system in the early 2000s through local park development, this foot-only route features gentle terrain with occasional stone steps built by the Civilian Conservation Corps, boardwalks over wetlands, and scenic overlooks of 80-foot limestone bluffs.^[23] Open daily with free public access, it includes amenities like parking, restrooms, and nearby fishing spots, making it ideal for casual walks, birdwatching, and picnics amid restored prairie and woodland habitats.^[24] Near Moab, Utah, the Devil's Staircase serves as a notorious extreme off-road obstacle within the Hell's Revenge 4x4 trail in the Sand Flats Recreation Area, managed by the Bureau of Land Management. This challenging section, consisting of steep sandstone ledges, slickrock fins, and high-consequence climbs, demands high-clearance four-wheel-drive vehicles with skilled drivers and is rated 10/10 in difficulty on enthusiast scales due to its technical demands and exposure.^[25] Originally closed for environmental reasons, a modified route was reopened in the early 2020s to balance recreation with resource protection, allowing access via guided tours or permitted personal vehicles while requiring adherence to tread-lightly principles in the desert ecosystem. The trail highlights Moab's iconic red rock formations and offers panoramic views of the Colorado River, attracting off-road enthusiasts year-round under seasonal permit requirements. The Devil's Stairs trail in eastern Idaho's Caribou-Targhee National Forest, near Driggs, presents an 8-mile out-and-back hike starting from Teton Canyon trailhead, rated moderate for its steady switchbacks ascending through ponderosa pine forests and open meadows to a scenic viewpoint at 8,500 feet. Documented in regional trail guides since 2015, this lesser-known path gains about 1,600 feet in elevation with rocky sections and wildflower displays in summer, remaining low-traffic compared to nearby Teton attractions due to its remote location.^[26] Accessible via free parking at the trailhead, it provides opportunities for wildlife observation, such as moose and deer, and serves as a gateway to longer backcountry routes without the crowds of Grand Teton National Park. Across these U.S. sites, the "Devil's Staircase" or "Stairs" nomenclature consistently evokes the steep, winding, and demanding topography that tests visitors, from forested switchbacks and river gorges to slickrock ascents, reflecting a shared cultural motif for challenging natural features in American landscapes.

Arts and entertainment

Literature

The Devil's Staircase is the title of a psychological thriller novel by Australian author Helen Fitzgerald, first published in 2009 by Polygon Books. The 224-page story centers on Bronny, a 19-year-old Australian backpacker who travels to London in search of adventure and sexual awakening, only to find herself squatting in a rundown Brixton house with a group of petty criminals and enigmatic figures, including a seductive but dangerous man and a mysterious woman named Sylvie. As Bronny navigates temptation and poor choices, the plot spirals into themes of murder, explicit sexuality, and the disorienting alienation of urban life abroad, culminating in a dark confrontation with hidden horrors in the house's basement.^[27] The novel has been praised for its innovative fusion of chick-lit narration with gritty noir crime elements, creating a tense, humorous, and unsettling read that critiques youthful naivety in a hostile environment. It was shortlisted for the 2010 Davitt Award for best adult crime novel by an Australian woman and nominated for the 2010 Spinetingler Award in the best novel category.^[28] Another literary work titled The Devil's Staircase is a 2014 non-fiction book by Australian magician Greg Chapman, self-published and distributed via Vanishing Inc. The 223-page hardcover delves deeply into advanced card magic, focusing on full-deck stacks, intricate sleights, historical context for gambling-themed effects, and practical routines for performers. Aimed at intermediate to expert magicians, it features over 50 black-and-white illustrations and Chapman's original contributions, including the effect "The Devil's Climb," a challenging routine involving progressive card revelations and subtle manipulations to simulate impossible controls. The book emphasizes technical precision and deceptive artifices, drawing on classic influences like Si Stebbins and Mnemonica stacks while innovating for modern close-up performances, and has been lauded in magic circles for its rigorous scholarship and practical utility.^[29] Both Fitzgerald's and Chapman's works under this title thematically evoke a perilous descent—moral and ethical in the novel's portrayal of Bronny's unraveling innocence amid London's underbelly, and technical in the book's exploration of the "hellish" complexities of flawless card handling and illusion craftsmanship.

Film and television

The 1964 South Korean film The Devil's Stairway (original title: Ma-ui Gyedan, also known as The Evil Stairs), directed by Lee Man-hee, is a black-and-white psychological thriller exploring themes of obsession, betrayal, and murder in a post-war urban setting.^[30] The 108-minute drama follows a businessman who attempts to end his affair with his possessive mistress to marry his fiancée, leading to a tense plot involving deception and a Diabolique-inspired twist, starring Kim Jin-kyu in the lead role alongside Jeong-suk Moon and Ae-ran Jeong. Noted for its atmospheric noir cinematography and strong performances, the film was restored and made available internationally through the Korean Film Archive, with screenings at festivals like the London Korean Film Festival in 2019.^[31] In television, the 2017 episode "The Devil's Staircase" from PBS's Infinite Series, hosted by mathematician Kelsey Houston-Edwards, is a 12-minute educational documentary segment that introduces the mathematical concept of the devil's staircase, illustrating Cantor's function through animations of the Cantor set and its counterintuitive properties like constant slope despite non-decreasing behavior. Aired on May 19, 2017, as part of the math history series produced by PBS Digital Studios, the episode aired on public television and was uploaded to YouTube, where it has garnered approximately 270,000 views as of 2025.^[32] The Korean film has been recognized as an influential work in Asian cinema, praised for its genre-blending suspense and contribution to early South Korean horror-thrillers, often screened in retrospectives of Lee Man-hee's oeuvre.^[33] The PBS episode, meanwhile, has found use in educational settings for teaching fractal geometry and real analysis, serving as an accessible entry point to advanced mathematical visualizations.^[34]

Music

The Devil's Staircase is a multinational eclectic progressive rock band formed in Chicago during the 2010s by musicians including bassist Luis Nasser, a physics professor at Columbia College Chicago.^[35] Now comprising six members scattered across the United States, Sweden, and Mexico—Tim McCaskey on acoustic guitar, Aaron Geller on electric guitar, Mattias Olsson on drums and Mellotron, Ramsés Luna on saxophone and electronics, and Edgar Arrellín Rosas on sound design—the group draws influences from King Crimson while incorporating themes from mathematics and science, such as fractals, chaos theory, and nonlinear dynamics.^[36]^[37] Their self-titled debut album, released independently on May 1, 2020, features eight instrumental tracks that sonically evoke these concepts, including "Rule 34" and "Room 101."^[36] As of 2025, the band was working on a follow-up album.^[38] In drum and bugle corps, the Santa Clara Vanguard presented a DCI World Class production titled "The Devil's Staircase" during their 2011 season, inspired by György Ligeti's Piano Étude No. 13 ("L'escalier du diable"), whose relentless ascending motifs mirror the constant but non-differentiable nature of the mathematical devil's staircase.^[39] Featuring brass, percussion, and synchronized visuals depicting a metaphorical mathematical ascent from hellish depths, the show was performed by about 150 members at more than 30 events across the DCI tour, earning a 6th-place finish with a score of 92.200 at the World Championship Finals in Indianapolis.^[40]^[41] The production is preserved in the Drum Corps International archives, with recordings available through official DCI channels. Other musical works bearing the title include Roth Händle Studios' recording of The Devil's Staircase band's 2020 self-titled album, an eclectic progressive rock release emphasizing intricate rhythms and thematic depth based on mathematical concepts.^[42] In contemporary classical music, Estonian-Canadian composer Udo Kasemets created Palestrina on Devil's Staircase around 1994, a chamber ensemble piece for two voices and strings that constructs musical forms based on Georg Cantor's fractal devil's staircase set, distributed by the Canadian Music Centre.^[43]^[44]

Other uses

Annual hillclimb race

The Devil's Staircase Pro Hillclimb is an annual professional motorcycle racing event organized by the Dayton Motorcycle Club and held at their facility on Powell Farm in Oregonia, Ohio.^[45]^[46] It serves as a key fixture in the AMA Pro Racing American Hillclimb series, featuring timed ascents up a challenging dirt hill known for its steep, multi-tiered terrain that tests rider technique, bike power, and traction control.^[47]^[45] The event emphasizes precision and skill over outright speed, with competitors launching from a near-vertical start to conquer the 470-foot course, often requiring multiple runs to determine class winners.^[48] The 76th edition took place on October 11–12, 2025, spanning two full days of competition that included the American Hillclimb East vs. West National Championship showdown.^[46]^[47] Riders competed in various professional classes such as 450 Pro, Premier, X-Climb, and Unlimited, with top seeds from regional series vying for national titles and overall supremacy.^[49]^[50] Gates opened at 9 a.m. each day, with racing commencing around 1 p.m., drawing dedicated enthusiasts to witness high-stakes climbs where elapsed times are measured to determine podium finishes.^[51] Established in 1950, the Devil's Staircase is one of the oldest continuously running hillclimb events in the United States, predating many modern motorsport series and maintaining its legacy through decades of evolution.^[45]^[52] The Dayton Motorcycle Club, founded in 1910, has hosted the event since at least 1973, preserving its status as a cornerstone of American motorcycle culture.^[53]^[54] It attracts elite riders from across the country, including champions like Jake Rassa in the East Premier class and Jake Anstett, who defended his title to win the overall national championship in 2025 (as of October 2025).^[55]^[56] In 2024, Jake Anstett claimed the top spot in the overall competition, with Lincoln Haugen winning the 50cc Invitational.^[56]^[57] It fosters a vibrant atmosphere with amateur divisions, youth invitational races (such as the 50cc class), and community traditions like on-site camping.^[58] Typically featuring around 32 professional entrants in the main classes plus additional amateur and invitational participants—totaling up to approximately 100 riders across all categories—the hillclimb underscores technical mastery on its unforgiving surface, where factors like tire choice and throttle control can make or break a run.^[59]^[49] Spectator admission is charged at $25–$30 per day or $50 for a weekend pass, supporting the event's operations while providing close-up views from designated seating areas parallel to the climbs.^[46]^[60] Rider entry details are managed through AMA Pro Racing protocols, with the event's significance amplified by its integration into the broader national series, where points contribute to annual championships.^[47]

In physics

In nonlinear dynamics, the devil's staircase describes a monotonically increasing function relating a dynamical parameter, such as the winding number, to a control parameter like driving frequency or amplitude, characterized by plateaus known as Arnold tongues and a fractal structure of steps arising from mode-locking phenomena.^[61] These structures emerge in systems exhibiting quasiperiodic motion transitioning to chaos, where the plateaus represent regions of phase locking and the steps reflect hierarchical frequency combinations.^[62] Unlike the purely mathematical Cantor function, which inspired the name due to its constant slope segments of zero measure, physical realizations feature finite-width steps modulated by system parameters, allowing observable entrainment ranges.^[61] A prominent example occurs in frequency-locked breathers, where numerical simulations of discrete nonlinear Schrödinger equations reveal a Farey tree hierarchy in the winding numbers, forming a devil's staircase with universal fractal scaling properties.^[63] This structure has been observed experimentally and theoretically in Josephson junctions under microwave irradiation, where current-voltage characteristics display subharmonic steps following continued fraction sequences, shifting with radiation amplitude.^[64] Similarly, in arrays of coupled oscillators or superconducting junctions, the staircase manifests as phase-locked states amid chaotic dynamics, with structured chaos unifying the hierarchical patterns.^[65] Applications span chaos theory, where the staircase elucidates transitions in driven systems; laser physics, as in modulated optical cavities exhibiting mode-locking; and superconductivity, particularly in superconductor-ferromagnet-superconductor junctions showing magnetization-coupled current steps.^[66] In fully frustrated superconducting arrays, subharmonic Shapiro steps contribute to the staircase, highlighting vortex interactions under combined currents.^[67] Recent simulations in 2025 have confirmed the universality of these structures in quantum systems, such as periodically driven time crystals in semiconductors, where frequency branches form devil's staircases during synchronization-to-chaos transitions, and in desynchronizing phase models displaying power-law behaviors.^[68]^[69]

References

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Cantor Function -- from Wolfram MathWorld
The Cantor function F(x) ; The Cantor function is a particular case of a devil's staircase (Devaney 1987, p. 110), and can be extended to a function F_q ; Chalice ...
[2]
[PDF] The Devil's Staircase - UCSD Math
A little additional thought shows that the limit function f is monotone increasing. It is called the Cantor Function or the Devil's staircase. Its graph is ...Missing: mathematical | Show results with:mathematical
[3]
The Cantor function - ScienceDirect
The graph of the Cantor function G. This graph is sometimes called “Devil's staircase”. We recall the definitions of the ternary Cantor function G and Cantor ...<|separator|>
[4]
The graph of the Cantor function G . This graph is sometimes called...
Cantor function G was defined in Cantor's paper [10] dated November 1883, the first known appearance of this function. In [10], Georg Cantor was working on ...
[5]
Cantor sets, the devil's staircase, and probability - The DO Loop
Jul 5, 2016 · The Cantor function is used in mathematics as a pathological example of a function that is constant almost everywhere yet somehow manages to climb upwards.
[6]
Devil's Staircase -- from Wolfram MathWorld
The Devil's staircase continuously maps the interval [0,1] onto [0,1], but is constant almost everywhere (ie, except on a Cantor set).
[7]
[PDF] A Note on the History of the Cantor Set and Cantor Function
After an exposition of the integration of discontinuous functions, Smith presented a method for constructing nowhere dense sets that were much more "substantial ...
[8]
[PDF] The Cantor function
Figure 1: The graph of the Cantor function G. This graph is sometimes called. ”Devil's staircase”. 1 Singularity, measurability and representability by ...
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https://doi.org/10.1016/j.exmath.2005.05.002
[10]
Devil's Staircase walk; Glen Coe - Discovering Britain
A short walk through scenic Glen Coe that explores why a road was built up a mountain and how it acquired a sinister name - Devil's Staircase.Missing: Major Caulfeild
[11]
Devil's Staircase - Roader's Digest: The SABRE Wiki
### Summary of Devil's Staircase as a Military Road in Scotland
[12]
This legendary trail is the best way to explore the Scottish Highlands
Nov 4, 2025 · Considered by many to be a moderate-to-difficult trail due to steep ascents and occasionally technical terrain, roughly 45,000 hikers attempt to ...
[13]
West Highland Way: Your Travel Guide - Wilderness Scotland
From the Kingshouse climb the trail known as the 'Devil's Staircase'. It is steep, eroded and with some switchbacks. The ascent is steady and constant. At the ...
[14]
Glencoe mountaineering | National Trust for Scotland
Aonach Eagach, (Notched Ridge), stretches 5.5 miles from the Devil's Staircase in the east to the Pap of Glencoe in the west and includes two Munros – Sgorr ...
[15]
The infamous Aonach Eagach Ridge – Glencoe - We Love Mountains
May 13, 2017 · Aonach Eagach is consistently ranked as the best ridge walk and scramble in the UK and in the top few across the world by National Geographic.
[16]
Haast Pass/Tioripatea Highway: Mount Aspiring National Park, Otago
This 140-km section of State Highway 6 is a spectacular and scenic road linking Wanaka and Haast. It reaches beyond wind-whipped Lakes Wanaka and Hāwea.Missing: SH6 descent
[17]
Opening of the Haast Road | West Coast region
Jan 29, 2020 · Norma Holyoake, wife of Prime Minister Keith Holyoake, cut the ribbon to open the final section of the Haast road at Knights Point on 6 November 1965.Missing: 1960s SH6
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https://fromasmallcity.nz/lake-wakatipu-and-the-devils-staircase/
[19]
Haast Pass in New Zealand | Things to see and do in New Zealand
Allow plenty of time to journey over Haast Pass to the coast. Short walks begin at the road's edge, leading you through rainforest to fabulous waterfalls.
[20]
Devil's Staircase Wilderness - Bureau of Land Management
This wilderness is part of a rugged and largely inaccessible area of old-growth forest in the Oregon Coast Range. Douglas-fir, western hemlock and cedar trees ...Missing: features | Show results with:features
[21]
Devil's Staircase, Wisconsin - 298 Reviews, Map | AllTrails
Rating 4.6 (298) Devil's Staircase is an out-and-back trail that covers 1.1 mi, with an elevation gain of 108 ft. It takes about 0.5–1 hr to complete. Out-and-back trails have ...
[22]
Devil's Staircase - Janesville Area Convention & Visitors Bureau
This mile and a half segment of the trail is for foot travel only and is family friendly. Parking, water, and restrooms are available in the park (seasonally).Missing: features | Show results with:features
[23]
The Devil's Staircase, Utah - GPS Trail Map, Conditions & Difficulty
The Devil's Staircase can be accessed by the following ride types: High-Clearance 4x4; SxS (60"); Dirt Bike. The Devil's Staircase Map.
[24]
The Devil's Stairs | Idaho Trail Junkie
Jul 10, 2015 · The Devil's Stairs is a hike of roughly 8 miles in-and-out. It's worth a visit, and chances are good that you'll have the place mostly to yourself.
[25]
Guardian review | Books
Apr 17, 2009 · The Devil's Staircase. Review: The Devil's Staircase by Helen Fitzgerald Although Bronny is an engaging character, this book has a remote ...<|control11|><|separator|>
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The Devil's Staircase–An Important New Card Magic Book
Nov 28, 2014 · ... card magic. Greg Chapman's new book is called The Devil's Staircase. If you do any kind of gambling material and know the difference between ...
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The Devil's Stairway (1964) - IMDb
Rating 6.6/10 (161) The Devil's Stairway: Directed by Lee Man-hee. With Kim Jin-kyu, Jeong-suk Moon, Ae-ran Jeong, Nam-hyeon Choi. A man wants to leave his mistress so he can ...
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The Evil Stairs - Wikipedia
The Evil Stairs (Korean: 마의 계단; also known as The Devil's Stairway) is a 1964 South Korean mystery film directed by Lee Man-hee. ... "마의 계단 The Evil ...
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The Devil's Stairway | KCCUK
Nov 13, 2019 · The Devil's Stairway. 23 Nov 2019 - 3:15pm. Venue: Edinburgh Filmhouse ... Lee Man-hee's atmospheric noir has two. Kim Jin-kyu often ...<|separator|>
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The Devil's Staircase | Infinite Series - YouTube
May 19, 2017 · Find out why Cantor's Function is nicknamed the Devil's Staircase. ... Kill the Mathematical Hydra | Infinite Series. PBS Infinite Series ...
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The Devil's Stairway - Movie review - easternKicks.com
Sep 15, 2023 · Lee Man-hee's 1964 horror-thriller The Devil's Stairway shines amongst its contemporaries with an intriguing storyline and strong cast.
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Pbs Video on Devil's Staircase - Sumant's 1 page of Math
May 27, 2017 · This is my first introduction to the Cantor Set and Cantor Function. The Cantor set is easily understood if one moves to the realm of base 3 ...
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2019 NJProghouse 20th Anniversary Hootenanny
The Devil's Staircase is known for their unique musical approach, which is inspired by math and physics. Columbia College Chicago physics professors Luis Nasser ...
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The Devil's Staircase | The Devil's Staircase
The Devil's Staircase is an international group of musicians who play rock inspired by math and science. But they don't play math rock. Ideas from nonlinear ...
[36]
THE DEVIL'S STAIRCASE discography and reviews - Prog Archives
THE DEVIL'S STAIRCASE is a Eclectic Prog / Progressive Rock artist from Multi-National. This page includes THE DEVIL'S STAIRCASE's : biography, ...
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A Look into the Past: "The Devil's Staircase" - General Effect Media
May 30, 2023 · This is how SCV's 2011 show, “The Devil's Staircase,” started. The show is inspired by György Ligeti's Piano Etude 13, which is often referred ...
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Recap analysis: DCI World Championship Finals
Aug 13, 2011 · Santa Clara Vanguard (6th-92.20) scored the same as in Semifinals, was 4th in Percussion and 6th in Visual Effect and all three performance ...
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Santa Clara Vanguard - DCX - Drum Corps Xperience
2011 36 scores. Highest Score 92.200. Final show in DCX Archives August 13, 2011 DCI World Championship Finals Indianapolis IN placed 6 with a score of 92.200.
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The Devils Staircase - Roth-händle - Bandcamp
Produced by The Devil's Staircase and Andy Tillotson. Please visit thedevilsstaircaseband.blogspot.com for complete musicology and discussion on the ...
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Palestrina on devil's staircase - Canadian Music Centre
Palestrina on devil's staircase. A construction based on the fractal set of George Cantor. Composer. Udo Kasemets. SKU: 13526. Call Number: MV 2203 K19pa
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Kasemets, Udo | Estonian Music Information Centre
... Canada Distributor: Canadian Music Centre. Palestrina on Devil's Staircase. A construction based on the fractal set of George Cantor Written to commemorate ...
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https://cmccanada.org/shop/13526/
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Devil's Staircase Pro Hillclimb
The 76th Annual Devil's Staircase Pro Hillclimb is coming up Oct 11th & 12th 2025. This event will be 2 days of non-stop hillclimb racing with the top 16 riders ...The Dayton Motorcycle Club · MX Pics · 2025 Event Schedule · Drag PicsMissing: history | Show results with:history
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AMA Pro Racing Announces 2025 American Hillclimb Schedule ...
October 11, 2025 - Devil's Staircase, Dayton Motorcycle Club, Oregonia, Ohio. October 12, 2025 - Devil's Staircase, Dayton Motorcycle Club, Oregonia, Ohio.<|control11|><|separator|>
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Devil's Staircase National Recap - American Hillclimb
The two clocked identical 6.077-second runs up the monstrous 470-foot Devil's Staircase on their first attempts. Rassa then posted a 6.449- ...Missing: miles | Show results with:miles
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The 76th Annual Devil's Staircase Pro Hillclimb is coming up Oct ...
Sep 3, 2025 · On Saturday Oct. 11th all 32 athletes from both sides of the country will compete on both hills in 450, X-climb and Premier class. Sunday they ...Missing: course length
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Devil's Staircase | American Hillclimb, sanctioned by AMA Pro Racing
The Devil's Staircase event is on October 8, 2023, at Dayton Motorcycle Club in Oregonia, OH, with 450 Pro, Premier, and Twins Hillclimb races.Missing: length | Show results with:length
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Devil's Staircase Pro Hillclimb - FlyBy Week
• Hillclimb Racing Starts 1pm. Directions: I-71 to exit 36, follow signs 2.5 ... Oregonia Ohio. Oregonia is 10 miles north of Kings Island. NO carry in ...Missing: course length gradient
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1967 BSA 650: Long-Running Vertical Weapon
making it one of the longest running motorcycle events in the country.
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My Town Monday: The Oregonia Hill Climbs | Jennette Marie Powell
Oct 10, 2011 · The Dayton Motorcycle Club has hosted the event since 1973. Local lore says that when the original owner passed away a few years ago, the ...
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Dayton Motorcycle Club - Facebook
The Dayton Motorcycle Club was founded in 1910, making it one of the oldest clubs in the country. It's purpose is to help promote the sport of motorcycling.
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American Hillclimb East - Facebook
Oct 5, 2025 · Congrats to the 2025 American Hillclimb Overall Champion of the East @jakerassa! He'll be representing the #1E plate at the long awaited ...
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The 2024 Devil's Staircase 50cc Invitational official overall results ...
Oct 14, 2024 · The 2024 Devil's Staircase 50cc Invitational official overall results. These kids put on one hell of show! Congratulations on the perfect ...
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American Hillclimb East on Instagram: "Here's how Lincoln Haugen ...
Oct 14, 2024 · Haugen earned himself the top spot at the 2024 Devil's Staircase ... Day 2 Results from the American Hillclimb National Championship at Devil's ...
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75th Annual Devil's Staircase Pro Hillclimb - Knuckle HQ
Gates open at 9am both days, admission is $25 on Saturday, $30 on Sunday, all admission tickets sold include a chance to win a new 2024 Buell Hammerhead 1190 as ...
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Devil's Staircase (2025) - All You Need to Know BEFORE You Go ...
Rating 5.0 (1) Jul 29, 2025 · Devil's Staircase is an annual event currently hosted by the Dayton Motorcycle Club. The American Hillclimb National Championship East/West ...
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Universal generation of devil's staircases near Hopf bifurcations via ...
Sep 10, 2020 · We evaluate the average phase velocity 〈 d ϕ d t 〉 or, in the case of the circle map, the rotation number ρ : = lim n → ∞ ϕ n n as a function ...
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Devil's Staircases in Continuous Systems with Modulated Forcing
May 3, 2019 · We show that, with harmonically modulated forcing, nonlinear oscillations close to a Hopf bifurcation generically reproduce the devil's ...
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Farey tree and devil's staircase of frequency-locked breathers in ...
Oct 2, 2022 · The winding numbers exhibit the hierarchy of the Farey tree and the structure of a devil's staircase. Numerical simulations of a discrete laser ...
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Devil's staircases and continued fractions in Josephson junctions
Dec 26, 2013 · We have found that the observed steps form very precisely continued fractions. Increase of the amplitude of the radiation shifts the devil's ...Article Text · INTRODUCTION · DEVIL'S STAIRCASE AND... · ANALYSIS OF THE...
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Structured chaos in a devil's staircase of the Josephson junction
Jul 22, 2014 · A more explicit dynamical manifestation of a devil's staircase has been observed in the motion of charged particles in two electrostatic waves.
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[1802.09212] Devil's Staircases in SFS Josephson Junctions - arXiv
Feb 26, 2018 · We study the effect of coupling between the superconducting current and magnetization in the superconductor/ferromagnet/superconductor Josephson junction.
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Devil's Staircase in a Fully Frustrated Superconducting Array
Remarkably, we find a series of subharmonic steps in addition to standard integer and half-integer giant Shapiro steps, leading to devil's staircase structure.
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Exploring nonlinear dynamics in periodically driven time crystal from ...
Mar 26, 2025 · Here, each branch in the frequency spectrum forms a devil's staircase. When an entrainment range is approached by going through an increasing ...
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Devil's Staircase and Power-Law Due to Desynchronization in a ...
A devil's staircase structure was observed in the desynchronization graph, which represents the time evolution of the phase difference.