Slope

Slope (pronounced slohp)

(1) To have or take an inclined or oblique direction or angle considered with reference to a vertical or horizontal plane; slant.

(2) To move at an inclination; obliquely to move.

(3) To direct at a slant or inclination; incline from the horizontal or vertical.

(4) To form or describe something with a slope or slant.

(5) A descriptor of ground or some aspect of the natural or built environment which has an incline, such as a hill.

(6) An inclination or slant, especially downward or upward; to lie or cause to lie at a slanting or oblique angle.

(7) Any deviation from the horizontal or vertical; an inclined surface.

(8) In mathematics, (1) the tangent of the angle between a given straight line and the x-axis of a system of Cartesian coordinates; (2) the derivative of the function whose graph is a given curve evaluated at a designated point.

(9) In slang, a disparaging and offensive term used to refer to a person of East Asian appearance.

(10) As slope off, a slang term (mostly UK, Australia & New Zealand) describing someone moving slowly, or furtively away, usually to avoid work or responsibility; a rare variation is “sloped-in”, used to describe to who arrive somewhere surreptitiously (those late for work etc).

(11) To follow an inclined course down a hillside (applied especially to natural features).

(12) In military use, as slope position, a drill command referring to the position in which a long-arm should be held.

1495–1505: From Middle English slope (go in an oblique direction), from the earlier adjectival meaning “slanting”, an aphetic variant of the Middle English aslope, from the Old English aslopen, past participle of āslūpan (to slip away), the construct being a- (away) + slupan (to slip).  From 1709 slope was used to mean "to be in a slanting position", the transitive sense "place in a slanting position" having been part of the language since circa 1600.  The derogatory slang meaning "oriental person" is attested from 1948.  Slopingly is an adverb, slopingness, sloper & slope are nouns, sloped & sloping are adjectives and sloped & sloping are verbs.

The slippery slope and the thin end of the wedge

Borrowed from political science, the terms “slippery slope” and “thin end (sometimes edge) of the wedge” are sometimes used interchangeably but, while both refer to similar processes, there are nuances which distinguish the two.  The idea is that a small, minor and perhaps innocuous change or innovation affecting something can trigger a chain of events which might result in unintended consequences; in that there are similarities with chaos theory but a slippery slope is much more specific and probably lineal.  There’s often overlap between the two and the distinctions are not always absolute but circumstances usually tends more to one than the other.  Both tend to be used in political discourse by extremists and fanatics and are often example of what is called the “slippery slope fallacy” such as the argument that if gay marriage is allowed, eventually the gay people will be allowed to marry their goats.  That argument really was raised by some who claimed it wasn't an extreme position to take, pointing out that two generations earlier, those who had been opposed to the decriminalization of homosexuality because it would put society on a slippery slope towards gay marriage had been accused of raising a “slippery slope fallacy”.  All things considered, goats seem safe.  The slippery slope is also a piece of imagery adopted sometimes by black-letter-law judges who oppose judicial activism.

The difference is essentially in the dynamics driving the process.  On a slippery slope, things happen because of the inherent inertia; the notion that of sitting on a slippery slope, the slide downhill an inevitable consequence of the physics of fluid dynamics and the force of gravity.  The downward path will happen naturally.  By contrast, the model of the thin end of the wedge is that of the wedge driven into the tree.  If left there nothing will happen but if the woodsman continues to hammer the edge into the trunk, at some point, the tree will fall.  An example of the thin end of the wedge was the deployment in the early 1960s by the Kennedy administration (1961-1963) of a small number of military advisors to support the government of South Vietnam.  It had never been intended that large-scale combat operations would be undertaken in Vietnam but, step-by-step, Washington increased the commitment.  An more familiar example of the slippery slope is to adopt a rigorous diet and then allow a weekly “cheat day”.  On the first cheat day, one might have just the one chocolate biscuit but the next week it’s a biscuit and a donut and within weeks it’s packets of both.

Anthony Trollope’s (1815-1882) 1857 novel Barchester Towers is set in a tranquil and leafy town but there are few novels which, without even a threat of bloodshed, so successfully and with such subtlety impart such feelings of incipient evil and a relentless undercurrent of dread.  In Victorian novels there were many characters of dubious virtue but few not actually homicidal managed to induce in readers such feelings of unease and distaste as the slimy Reverend Obadiah Slope.

Although said to be “tall and not ill made”, Slope was “saucer-eyed”, his hair “lank, and of a dull pale reddish hue… formed into three straight lumpy masses, each brushed with admirable precision, and cemented with much grease.”  His face, “perhaps a little redder” than his hair, not unlike beef “of a bad quality”, a “redeeming feature” his nose which was “pronounced, straight, and well-formed” although marred still by “a somewhat spongy, porous appearance, as though it had been cleverly formed out of red colored cork".  The description of the nose is not without significance for it had by some been asserted that he was of lineal descent from Dr Slop, “that eminent physician who assisted at the birth of Mr Tristram Shandy, and that in early years he added an ‘e’ to his name, for the sake of euphony.”  Shandy, the eponymous character from Laurence Sterne's (1713-1768) nine volume work (1759-1757), The Life and Opinions of Tristram Shandy, Gentleman, was birthed by the “man-midwife” Dr Slop, who squashes Tristram's nose with forceps as he yanks the baby out.

Slope was the domestic chaplain to Bishop Proudie at Barchester and began as a protégé of the Bishop’s wife (a truly ghastly woman) but later became her enemy as he attempted when he endeavored to wrest the control of the diocese from her hands by becoming an eminence (pâle) rouge, a kind of vicarage Richelieu controlling the Bishop.  The plots and schemes of the calculating chaplain, conducted with much obsequiousness, play out in the novel as a part of a struggle between those of the high church and the evangelicals, a struggle not resolved to this day.  The Trollope aficionados don’t regard Barchester Towers as his best work but few deny it’s one of the most enjoyable and anyone who wishes to sample Trollope should start here.

On the slopes: Lindsay Lohan in Gstaad, Switzerland, 2016.  The experience on skis may have come in handy in 2022 during filming for Netflix's Falling for Christmas.

Climate

Climate (pronounced klahy-mit)

(1) The composite or generally prevailing weather conditions of a region, as temperature, air pressure, humidity, precipitation, sunshine, cloudiness, and winds, throughout the year, averaged over a series of years (decades, centuries or other epochs).

(2) A region or area characterized by a given climate (often as climatic zone or with modifiers such as cold-climate, sub-climate, micro-climate, dry-climate et al).

(3) The prevailing attitudes, standards, or environmental conditions of a group, period, or place (used in politics, sociology, economics etc in the sense of mood, atmosphere, spirit, tone, temper etc).

(4) An area of the earth's surface between two parallels of latitude (obsolete).

(5) A region of the Earth (obsolete).

(6) In contemporary slang, a clipping of “climate change” and frequent verbal (especially oral) shorthand.

1350–1400: From the Middle English climat, from the Old French climat (region, part of the earth), from the Late Latin clīmat- (stem of clīma, from the Ancient Greek κλῐ́νω (klínō) (I slope, incline)), from klīmat- (stem of klī́ma (latitude, slope, region (literally “inclination”), the notion being “slope of the earth from equator to pole”), the construct being klī- (akin to klī́nein (to slope, lean)) + -ma (the noun suffix), ultimately from a suffixed form of the primitive Indo-European root klei- (to lean).  The adjective climatic is sometimes confused with climactic.  Climatic is used of the climate; climactic describe something which forms a climax.  Climate is a noun & verb and climatic, climatical & climatal are adjectives and climatically is an adverb; the noun plural is climates.

The original meaning in the mid-fourteenth century was purely geographic, the sense being horizontal zone of the earth's surface measured by lines parallel to the equator (ie latitude).  From antiquity, geographers and cartographers divided earth into zones determined by the angle of the sun on the slope of the surface and the length of daylight.  Not knowing what lay beyond reported (and not always reliable) observations, these zones varied greatly.  Some calculated variously 24 or 30 climates between Meroe on the upper Nile (in modern-day Sudan) and the mythical Riphaean Mountains which thought to be the Arctic’s boundary, systems in which as one traced the map north, the climate changed at the point where daylight was a half hour longer or shorter (according to season) than the starting point.  Others thought cosmically and listed seven (one for each known planet) or twelve (tied to the signs of the zodiac).

Being a measure of latitude, there was obviously a tendency for the climates to correlate with temperature, something noticed as early as the late-fourteenth century and, being practical rather than abstract, the word came rapidly to be used in the sense of “distinct zones of the earth's surface associated with changes in weather”.  This meaning shift evolved by the early seventeenth century to make climate understood as “combined results of weather associated with a region, the characteristics of a country or region with reference to variation of heat, cold, humidity, rainfall, wind etc.  The figurative use (the collective mental or moral atmosphere of a group or whole society) dates from the 1660s.  The difference between climate and “the weather” is that climate is a summary of the long-term manifestations of weather and other atmospheric conditions (which can be applied to defined zones or globally).  The modern practice is that climate is a statistical summary of a representative period and that’s usually generational (ie 25-30 years) but climate modelers using the geological record and other sources (trees, ice cores etc) can construct longer epochs in centuries (eg the ice ages) or even millions of years, the global climate during the age of the dinosaurs often summed up as “warm & wet”.

Lindsay Lohan pondering climate change.

To acclimatise (or acclimatize) is to acclimate (become accustomed to) one’s self to the weather conditions prevalent where one find’s one’s self, a term used also figuratively in the sense of adjusting tastes or expectations to different cultures.  The use of modifiers is common and can be meteorological (warm-climate, dry-climate, chilly climate, temperate-climate etc) or figurative (hostile climate, sceptical climate etc).  In the age of climate-concern, there’s been a growth of such use including climate canary, climate control, climate denial, climate denier, climate despair, climate emergency, climate finance, climate doom, climate system, climate-speak and, of course, climate change.  Indeed, the very phrase “climate change” is a construct which has come widely to be used because it references easily understood consequences rather than some abstract process.  The possible climatic implications of man-made atmospheric CO₂ emissions was discussed as early as 1896 by Swedish chemist & Nobel laureate Svante Arrhenius (1859-1927) in a paper called On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground, his conclusion being that the burning of fossil fuels could lead to increased concentrations of carbon dioxide in the atmosphere, leading a global increase in average temperatures.  In the scientific community, his ideas for decades remained controversial and in the wider community, barely noted and it wasn’t until the 1950s when researchers, taking advantage of new techniques made possible by the vastly increased quantity of data which became available because of wartime advances and a vast expansion of meteorological and atmospheric measuring, began serious study of the study the potential impact of human activities on the Earth's climate.

Between the 1970s and the early twenty-first century, a consensus built in mainstream science that human activities (especially the burning of fossil fuels), were contributing to global warming and other climatic changes.  The early phrase popular in the discussion, at that stage conducted mostly among scientists and environmentalists (although the fossil fuel industry quickly mounted counter campaigns, both in public relations activities and the traditional buying-off of politicians) was the “greenhouse effect” but greenhouses, associated with positive things like grapes & tomatoes didn’t have the desired worrying effect so the terminology in the 1990s shifted to the consequences.  “Global warming” replaced “greenhouse effect” in consciousness-raising but it proved vulnerable to attacks from industry which claimed every unseasonal cold-spell disproved the thesis, thus the use of “climate change”.  That has proved resistant to challenge and even in the West the matter is now being taken seriously, as the consequences of climate change begin to affect to rich nations of the global north, destructive heat, fires, hurricanes, floods and tropical diseases no longer ignored as largely they were when those who suffered were mostly Africans, Arabs, Asians and animals.

Psychiatrists & psychologists find the distinction between climate and weather a helpful metaphor when explaining personality.  The term personality denotes the enduring pattern of a patient’s life and can thus be thought their individual climate.  The weather which produces that climate can be good (happy events, a successful marriage, a fulfilling career), bad (a dull job, habitually sore feet) or severe (cancer, divorce) and in some cases a patient’s reaction to these event can result is what is diagnosed as a personality disorder (PD).  In the West, psychiatry is thought one of medicine’s more recent disciplines and in some structural senses that’s probably true but the understanding of the human personality is ancient.  Writing in 45 BC, the Roman statesman & scholar Cicero (106-43 BC) observed that in men existed “…all kinds of depravity and perversity” and that “There are more disorders of the mind than of the body and they are of a more dangerous nature for the mind, when disordered… can neither bear nor endure anything and is under the perpetual influence of desires”.  Cicero went further, noting the depravities of men were deep seated and either caused or exacerbated by environment in which men lived and what afflicted one could also poison the whole for when an entire society “declare unanimously for what is wrong, then we are altogether overwhelmed”.  PD writ large: Each individual an element of the weather and the body politic the climate.

La Mort de Cicéron (The Death of Cicero (circa 1635), oil on canvas by François Perrier (1590–1650), Bad Homburg, Staatliche.

Within two years of writing those words Cicero would be dead and the circumstances of his execution would provide probably several case studies in PD but he had few reasons to complain; he may have lived by the pen but he ruled by the sword, for centuries remembered fondly by scholars of Antiquity and, perhaps less helpfully for his reputation, of late by Ted Cruz (b 1970; US senator (Republican, Texas) since 2013).  Knowledge of PD then was not novel when in 1980 when first it appeared in the third edition (DSM-III) of the American Psychiatric Association's (APA) Diagnostic and Statistical Manual of Mental Disorders, organized on a categorical basis with patients classified into one of several distinct personality disorder types.  This system was revised in the DSM-IV (1994) and again in the DSM-5 (2013), the latter adopting a hybrid categorical-dimensional approach to PD, a condition it notes affects at least two of a patient’s (1) way of thinking about oneself and others, (2) way of responding emotionally, (3) way of relating to other people and (4) way of controlling one’s behavior.  In the DSM-5-TR (2022), the types of PD are listed as:

Antisocial personality disorder: A pattern of disregarding or violating the rights of others.  A person with antisocial personality disorder may not conform to social norms, may repeatedly lie or deceive others, or may act impulsively.

Avoidant personality disorder: A pattern of extreme shyness, feelings of inadequacy, and extreme sensitivity to criticism.  Patients with avoidant personality disorder may be unwilling to get involved with others unless they are certain of being liked, be preoccupied with being criticized or rejected, or may view themselves as not being good enough or socially inept.

Borderline personality disorder: A pattern of instability in personal relationships, intense emotions, poor self-image and impulsivity.  A patient with borderline personality disorder may go to great lengths to avoid being abandoned, have repeated suicide attempts, display inappropriate intense anger, or have ongoing feelings of emptiness.

Dependent personality disorder: A pattern of needing to be taken care of and submissive (clingy the modern term) behavior.  Patients with dependent personality disorder may have difficulty making daily decisions without reassurance from others or may feel uncomfortable or helpless when alone because of fear of inability to take care of themselves.

Histrionic personality disorder: A pattern of excessive emotion and attention-seeking. Patients with HPD may be uncomfortable when they are not the center of attention and may use physical appearance to draw attention to themselves or have rapidly shifting or exaggerated emotions.

Narcissistic personality disorder: A pattern of need for admiration and lack of empathy for others.  A patient with NPD may have a grandiose sense of self-importance, a sense of entitlement, take advantage of others or lack empathy.

Obsessive-compulsive personality disorder: A pattern of preoccupation with orderliness, perfection and control.  A patient with OCPD (distinct from Obsessive-compulsive disorder (OCD)) may be overly focused on details or schedules, may work excessively, not allowing time for leisure or friends, or may be inflexible in their morality and values.

Paranoid personality disorder: A pattern of being suspicious of others and seeing them as mean or spiteful.  Patients with paranoid personality disorder often assume people will harm or deceive them and refuse to confide in others or become close to them.

Schizoid personality disorder: Being detached from social relationships and expressing little emotion.  A patient with schizoid personality disorder typically does not seek close relationships, chooses to be alone and appears to care little about praise or criticism from others.

Schizotypal personality disorder: A pattern of being very uncomfortable in close relationships, having distorted thinking and eccentric behavior. A patient with schizotypal personality disorder may have odd beliefs or odd or peculiar behavior or speech or may have excessive social anxiety.

Tumblehome

Tumblehome (pronounced tuhm-buhl-hohm)

(1) In naval architecture, an inward and upward slope of the middle body of a vessel; of the sides of a ship: To incline or slope inwards, to contract above the point of extreme breadth.

(2) A conceptually similar shape applied, in reverse, to the upper body of an automobile.

1828: A compound word, tumble + home.  Tumble was from the Middle English tumblen (to fall over and over again, tumble), frequentative of the Middle English tumben (to fall, leap, dance), from the Old English tumbian, from the Proto-Germanic tūmōną (to turn, rotate).  It was cognate with the Middle Dutch tumelen and the Middle Low German tumelen & tummelen.  Home was from the Middle English home, hom, hoom & ham, from the Old English hām (village, hamlet, manor, estate, home, dwelling, house, region, country), from the Proto-West Germanic haim, from the Proto-Germanic haimaz (home, village), from the Proto-Indo-European tḱóymos (village, home), from the root tḱey-.  The (rare and probably extinct) alternative spelling is tumble home.  Tumblehome is a noun; the noun plural is tumblehomes.

The meaning of the word tumblehome has been well understood from its first appearance in the early nineteenth century but the origin has never been obvious.  Shipbuilders had for centuries been using variations of the design for a number of reasons but the first known instance of the word dates only from 1828 and then without explanation, suggesting the term may already have been in common use, at least within the industry.  An 1848 reference from a shipwright does however hint at some sense of novelty, noting “… the upper works usually incline towards the middle line, or as it is termed “tumble home”.  The word “tumble” to refer to the sides of ships appears to have been used at least as early as 1687 but the compound tumblehome seems not to have emerged for another hundred and fifty-odd years.  The idea always summoned was of the imagery of the sides of a ship “tumbling down” the slope created but why “home” was added remains a mystery, the assumption being it was based either on (1) an association with certain domestic architectural styles of the time (2) the romantic notion of the sea, to which the tumblehome falls, being "home" for sailors or  (3) the idea of a dilapidated house in the throes of "tumbling down", fallen bits an pieces accumulating at the bottom.

Big ships and fast cars

In automotive design (upper), the term is applied when the width of the cabin (cockpit or glasshouse) reduces as the height rises.  Although curved glass in the side windows of cars began widely to be adopted in the mid-1960s, thus creating a mild tumblehome effect, the term is applied only when it is sufficiently severe to be apparent to casual viewers.

In naval architecture (lower), the geometry is reversed, a tumblehome define as a hull which flares out as the sides approach the waterline.  Although in some vessels, the effect is barely detectable by the naked eye, it’s a technical term and applies to all hulls which dimensionally qualify.  The opposite, the classic shape for ships’ hull, is called the flare.

USS Brooklyn, 1896.

Tumblehome, unless taken to extremes, was functional in that it improved stability in warships under sail; sailing ships heel (they tend to lean over when moving) and tumblehome reduced this.  At the time, the biggest contributor to a warship’s mass on the upper decks was the guns and a tumblehome design, moving the centre of gravity lower, allowed armament to be maintained or even increased without further loss of stability.  Additionally, there was the benefit of making it harder for boarding parties to climb aboard.  In commercial shipping, vessels were long taxed on the basis of the square footage of a ship’s deck and fat ships with a pronounced tumble carrying the same freight but taxed less, were attractive.  Government fiscal policy thus influenced and distorted design and engineering principles in the same way tax arrangements of windows affected architecture and those on cylinder bores (adversely) affected engine design.

Lamborghini LP500 Countach prototype, 1971.

The Countach had one of the most extreme implementations, the angle meaning it was possible for only part of the side-window to be lowered but at least the Italians were more thoughtful than the Germans; in 1954, facing a similar challenge with the side-glass on the 300 SL (W198 1954-1957) gullwing, Mercedes-Benz simply fixed the panes, ventilation provided only by small quarterlights.  Neither flow-through ventilation or air-conditioning was available so driving in a gullwing could be hot and sticky experience and there's a reason they're sometimes seen being driven (at low speeds and not on public roads) with at least one door open. .  The tumblehome is used by high-performance cars because of the aerodynamic advantages it confers, reducing frontal area an allowing the curve of the greenhouse to be optimized for air-flow, lowering resistance.  Because of great advances made during the late twentieth century, refinements to tumblehomes no longer deliver the 3-5% improvements in a drag coefficient (CD) which once was possible, engineers now pursuing factional gains.  The origins in cars however lay in the quest for more interior space and for mass-market vehicles, bulging out the sides gained the odd vital inch and the technique, combined with curved side glass, has become almost universal although there has been the odd deviation.  Stylists are predicting tumblehomes are likely to become more exaggerated as sides need to be bulkier to meet more rigorous side-impact regulations and roof-lines are lowered slightly in the quest to reduce drag.

Lindsay Lohan in tumblehome blonde wig.

What professional hair stylists call “the tumblehome” is a triangulated shaping which is most cases can’t be achieved without an expert application of product and when sported by models on photo-shoots, it’s common for the angles and an illusion of volume to be achieved with engineering no more complex than a sheet of cardboard (cut to suit) being attached with hairclips to the back of the head.  The look can however be achieved with synthetics which can be persuaded sustainably to behave in a way human hair naturally resists and Lady Gaga (b 1986) made a tumblehome wig a signature feature of her “Fame Monster” period (2009-2010).  With natural hair, a tumblehome with hair a little shorter than that of Lady Gaga’s wig is sometimes technically achievable given the right hair and a generous use of product the sideways projection would be noticeably less.

Lady gaga in Fame Monster mode.

The tumblehome style with the exaggerated elongations al la Gaga is rarely seen and usually represents a lot of work.  However, many take about as much effort to avoid the similar geometry of the “pyramid head”: a triangular shape with a flat crown area which flares to a wide bushy shape at the ends.  A function of length and weight for those with curly hair, pyramids happen usually when the strands are of almost uniform length and the curls tend to “stack”, the weight meaning the roots sit flattest on the scalp while towards the ends where the effective volume (hair + space) is greatest, the curled strands move sideways, unlike the behavior of straight hair which is purely downwards.

Lindsay Lohan with pyramid head, Saturday Night Live, 2004.

Stylists recommend layers as the best tactic to minimize the triangulation, the strategy essentially to create longer, diagonal layers to frame the face, meaning the remaining curls “sit into each other”.  What this does is simply physics, the layering on the surface reducing the weight, increasing the percentage of the volume on the crown area and although some are resistant, the best results will probably be achieved if the hair is cut dry because it will be presented at its natural weight.  When wet, the moisture content will disguise the extent of the left-right movement and exaggerate the up-down.  The shorter the layers of course the more effective the amelioration but this can be too radical for some so clients need to be turned into realists.

Ogee

Ogee (pronounced oh-jee)

(1) A double curve, resembling the letter S, consisting of two arcs that curve in opposite directions so that the ends are parallel.   An ogee is a kind of sigmoid curve, technically formed by the union of a concave and a convex line.

(2) In architecture, a molding with such a curve for a profile; cyma, also called gula or talon.

(3) In fluid mechanics, a specific aerodynamic profile.

(4) In mathematics, an inflection point.

(5) In cosmetic surgery, a desired shape for the curve of the malar or cheekbone prominence transitioning into the mid-cheek hollow.

(6) In distillation, the bubble-shaped chamber of a pot still connecting the swan neck to the pot which allows distillate to expand, condense and fall back into the pot.

1275-1325: From the Middle English ogeus (plural oggez), ogeve, egeve & egeove, variants by assimilation from the French oggif (diagonal rib of a vault) from the Old French œgive, ogival, ogive & augive, from the Late Latin augiva of uncertain origin.  In the Late Latin there was ogis (a support, prop), thought derived from the Classical Latin augeō (to increase, strengthen) from which Spanish gained auge (highest point of power or fortune; apogee); a doublet of ogive.  The use in architecture to describe regular & irregular S-shaped moldings began in the 1670s, supplanting the earlier augive, from the Late Latin augiva of uncertain origin but perhaps related to the Latin via (way; road).  In the Middle English, the late thirteenth century ogif was "a stone for the diagonal rib of a vault", derived from the French and the Medieval Latin ogiva.

An Ogee arch.

An ogee is a curved shape vaguely like an “S”, consisting of two arcs that curve in opposite senses, so that the ends are tangential.  In architecture, it’s used to describe a molding with a profile made by a lower concave arc flowing into a convex arc.  Used first in Persian and Greek architecture from Antiquity, it’s most familiar now as a part of the Gothic style.  Because the upper curves of the ogee arch are reversed, it’s not suitable where the load-bearing rates are high and it’s thus used mostly in self-supporting structures exposed to their own weight and not subject to high external forces.

An inflection point.

In mathematics, an ogee is an inflection point, a point on a curve at which the sign of the curvature (ie the concavity) changes; inflection points may be stationary but are neither local maxima nor local minima.  Inflection Points merely mark the point on the curve where it changes from concave upward to concave downward (or vice versa), it is not a directional measure.  Calculus is needed to find where a curve goes from concave upward to concave downward and for this mathematicians use derivatives, one which determines the slope and a second which defines whether the slope increases or decreases.

The cosmetic surgeon's template.

When cosmetic surgeons speak of the ogee curve, they’re referring to the elongated S curve which, when looking at a patient’s face at a three-quarter aspect, describes the curve from the eye through the cheek down to below the cheek (technically the malar or cheekbone prominence transitioning into the mid-cheek hollow).  As people age, the face loses volume in our face and the ogee curve flattens out. With facelifts and related procedures, the cosmetic surgeon aims to create an ogee curve, restoring the chiseled refinement of youth.

Lindsay Lohan, Cosmopolitan, October 2022 edition (photographs by Ellen Von Unwerth (b 1954)).  The photographs illustrate the ideal ogee curve in the facial structure and Ms Lohan shows how a model's pout can be used to optimize the effect.  The car is a second generation AMC Javelin (1971-1974).

Shapes derived from the ogee curve are popular design motifs used for just about everything where they can be applied but it's most associated with large, flat surfaces because it's in such spaces the geometric repetition can be most effective, thus the prevalence in fabrics, wallpaper and floor-coverings.  The mind visualizes the shape as something like a two-headed onion and it's probably the association with the architectural style so often seen on mosques or other structures in the Arab world that in there's presumed to be some Islamic influence but ogee long pre-dates Islam.  There are few conventions of use except that the designs tend to be displayed in either a horizontal or vertical aspect; diagonal deployment is rare.

In fluid mechanics, the ogee describes the shape used to create certain aerodynamic profiles, the classic example being the delta wing of the Concorde airplane.  The ogee-type wing used on the Concorde one flavor of the compound delta (sometimes called double delta or cranked arrow) where the leading edge is not straight.  Typically the inboard section has increased sweepback, creating a controlled high-lift vortex without the need for a foreplane and the shape of the sweepback is defined by the parameters of the ogee.  On the Concorde, the two sections and cropped wingtip merge into a smooth ogee curve.

Canthus

Canthus (pronounced kan-thuhs)

The angle or corner on each side of the eye, formed by the natural junction of the upper and lower lids; there are two canthi on each eye: the medial canthus (closer to the nose) and the lateral canthus (closer to the ear).

1640–1650: From Ancient Greek κανθός (kanthós) (corner of the eye) (and also an alternative spelling of cantus (in music, sung, recited, sounded, blew, chanted etc)), which became conflated the New Latin canthus, from the Classical Latin cantus (the (iron) rim of a wheel)).  The term describing the “iron rim of a wheel” was ultimately of Gaulish origin, from the Proto-Celtic kantos (corner, rim) and related to the Breton kant (circle), the Old Irish cétad (round seat) and the Welsh cant (rim, edge).  The Greek form was borrowed by Latin as canthus and with that spelling it entered English.  In the medieval way of such things, canthus and cantus became conflated, possibly under the influence or regional variations in pronunciation but some etymologists have noted there was tendency among some scribes and scholars to favor longer Latin forms, for whatever reason more letters being thought better than fewer.  The most familiar descendent in music is the canto (a description of a form of division in composition with a surprisingly wide range of application).  Canthus is a noun and canthal is an adjective; the noun plural is canthi (pronounced kan-thahy).

One word in English which has long puzzled etymologists is the late fourteenth century cant (slope, slant) which appeared first in Scottish texts, apparently with the sense “edge, brink”.  All dictionaries list it as being of uncertain origin and the Oxford English Dictionary (OED) notes words identical in form and corresponding in sense are found in many languages including those from Teutonic, Slavonic, Romanic & Celtic traditions.  Rare in English prior to the early seventeenth century, the meaning “slope, slanting or tilting position” had been adopted by at least 1847 and may long have been in oral use.  The speculation about the origin has included (1) the Old North French cant (corner) which may be related to the Middle Low German kante or the Middle Dutch kant, (2) the New Latin canthus, from the Classical Latin cantus (the (iron) rim of a wheel), (3) the Russian kutu (corner) and (4) the Ancient Greek κανθός (kanthós) (corner of the eye).  To all of these there are objections are the source remains thus uncertain.

The metrics of the attractiveness of women

PinkMirror is a web app which helps users optimize their facial aesthetics, using an artificial intelligence (AI) engine to deconstruct the individual components an observer’s brain interprets as a whole.  Because a face is for these purposes a collection of dimensions & curves with certain critical angles determined by describing an arc between two points, it means things can be reduced to metrics, and the interaction of these numbers can used to create a measure of attractiveness.

Positive, (left), neutral (centre) & negative (right) eye canthal tilt.

Perhaps the most interesting example of the components is the eye canthal tilt, a positive tilt regarded as more attractive than a negative.  The eye canthal tilt is the angle between the internal corner of the eyes (medial canthus) and the external corner of the eyes (lateral canthus) and is a critical measure of periorbital (of, pertaining to all which exists in the space surrounding the orbit of the eyes (including skin, eyelashes & eyebrows) aesthetics.  The eye canthal tilt can be negative, neutral, or positive and is defined thus:

Positive: Medial canthus tilt between +5 and +8^o below the lateral canthus.

Neutral: Medial canthus and lateral canthus are in a horizontal line.

Negative: Medial canthus tilt between -5 to -8^o below the lateral canthus.

Pinkmirror cites academic research which confirms a positive canthal tilt is a “power cue” for female facial attractiveness and while it’s speculative, a possible explanation for this offered by the researchers was linked to (1) palpebral (of, pertaining to, or located on or near the eyelids.) fissure inclination being steeper in children than adults (classifying it thus a neonatal feature) and (2) it developing into something steeper still in females than males after puberty (thus becoming a sexually dimorphic feature).  Pinkmirror notes also that natural selection seems to be operating to support the idea, data from Johns Hopkins Hospital finding that in women, the intercanthal axis averages +4.1 mm (.16 of an inch) or +4^o, the supposition being that women with the advantage of a positive medial canthus tilt are found more attractive so attract more mates, leading to a higher degree of procreation, this fecundity meaning the genetic trait producing the characteristic feature is more frequently seen in the population.  Cosmetic surgeons add another layer to the understanding, explaining the canthal tilt is one of the marker’s of aging, a positive tilt exuding youth, health, and exuberance where as a line tending beyond the negative is associated with aging, this actually literally product of natural processes, the soft tissue gradually descending under the effect of gravity, as aspect of Vogue magazine’s definition of the aging process: “Everything gets bigger, hairier & lower”.

With people, medial canthus tilt is thus an interaction of (1) the roll of the genetic dice and (2) the cosmetic surgeon’s scalpel.  With manufactured items however, designers have some scope to anthropomorphize objects and few visages are as obviously related to a human’s eyes than the headlamps on a car.

The positive, neutral & negative: 1965 Gordon-Keeble GK-1 (left), 1958 Edsel Corsair Hardtop (centre) & 1970 Maserati Ghibli Roadster (right).

When headlamps were almost universally separate circular devices, the creation of a medial canthus tilt really became possible in the mid-1950s after dual units were first made lawful in the US and then rapidly became fashionable.  Overwhelmingly, the designers seemed to prefer the neutral and where a positive tilt was use, it was exaggerated well beyond that found in humans.  Instances of the negative were rare, which would seem to support the findings of attractiveness in humans but they were sometimes seen when hidden headlamps were used and there they were necessitate by the form of the leading edge under which they sat.  The suspicion is that designers found a negative slant acceptable if usually they were hidden from view.

2005 Porsche 911 Turbo S (996) (left), 2016 Ford (Australia) Falcon XR8 (FG) (centre) & 2000 Ferrari 550 Maranello.

As the interest in aerodynamics grew and there were advances in shaping glass and plastic economically to render compound shapes, headlights ceased to be merely round (though rectilinear shapes did start to appear in the 1960s) and took on abstract forms.  The demands of aesthetics however didn’t change and designers tended still to neutral or positive tilts.  Care needed still to be taken however, the derided “poached egg” shape on the 996 generation of the Porsche 911 (1997-2006) not popular with the obsessives who buy the things, their view being each update should remain as devoted to the original (1963) lines as themselves.  One of the closest to a flirtation with a negative tilt showed up on the Ferrari 550 Maranello (1996-2001) and the factory hasn’t repeated the experiment.

Deconstructing Lindsay Lohan

The Pinkmirror app exists to quantify one’s degree of attractiveness.  It’s wholly based on specific dimension and thus as piece of math, is not influenced by skin tone although presumably, its parameters are defined by the (white) western model of what constitutes attractiveness.  Users should therefore work within those limitations but the model would be adaptable, presumably not to the point of being truly cross-cultural but specifics forks could certainly be created to suit any dimensional differences between ethnicities.  Using an industry standard known as the Photographic Canthal Index (PCI), one’s place on Pinkmirror’s index of attractiveness is determined by the interplay of (1) Nose width, (2) Bi-temporal to bi-zygomatic ratio, (3) chin length, (4) chin angle, (5) lower-lip height & (6) eye height.

Lindsay Lohan scored an 8.5 (out of 10), was rated as “beautiful” and found to be “very feminine, with great features of sexual dimorphism”, scoring highly in all facets except lower lip height and eye height.  Her face shape is the heart, distinguished by a broad forehead and cheekbones, narrowing in the lines of down to the jaw-line, culminating in a cute pointy chin.  Pinkmirror say the most attractive face shape for women has been found to be the triangle, scoring about the same as the oval while the heart, round, diamond, rectangle and square are also attractive to a lesser degree.  Within the app, pears and oblongs are described as “not typically seen as attractive” and while the word “ugly” isn’t used, for the unfortunate pears and oblongs, that would seem the implication.

Perimeter & Parameter

Perimeter (pronounced puh-rim-i-ter)

(1) A line bounding or marking off an area; any boundary around.

(2) The outermost limits.

(3) In geometry, the border or outer boundary of a two-dimensional figure (the sum of the lengths of the segments that form the sides of a polygon.

(4) The total length of such line; the total length of any such closed curve, such as the circumference of a circle.

(5) In military jargon a fortified boundary that protects a position.

(6) In clinical ophthalmology, an instrument for determining the peripheral field of vision.

(7) In basketball, a semicircular line on a basketball court surrounding the basket, outside of which field goals are worth three points rather than two (also called three-point line).

(8) The area outside this line (often used attributively).

1585–1595: From the French périmètre (circumference, outer boundary, or border of a figure or surface), from the feminine Latin form perimetros, from the neuter Greek perímetron (circumference), the construct being peri- (around; about) + -meter from metron (measure), from the primitive Indo-European root me- (to measure).  The military sense of “boundary of a defended position” is said by some sources to have come into use only by 1943 despite the tactic being probably the second oldest military procedure still in use (the attack presumably the first).  Whether coincidental or not, the ultimate failure of perimeter defense was what finally led to the success of the Soviet offensive against the Nazi Sixth Army in Stalingrad (now Volgagrad) in 1943.  The technical terms created by the use of perimeter as a modifier include perimeter check (a patrol which checks to ensure a defensive perimeter remains in place) & perimeter fence.  Perimeter & perimetry are nouns, perimetral, perimetric & perimetrical are adjectives and perimetrically is an adverb; the noun plural is perimeters.

Parameter (pronounced puh-ram-uh-tuhr (U) or puh-ram-i-ter (non-U)

(1) In mathematics, a constant or variable term in a function that determines the specific form of the function but not its general nature, as a in f(x) = ax, where a determines only the slope of the line described by f(x).  (A value kept constant during an experiment, equation, calculation or similar, but varied over other versions of the experiment, equation, calculation etc).

(2) In mathematics, one of the independent variables in a set of parametric equations.

(3) In geometry, in the ellipse and hyperbola, a third proportional to any diameter and its conjugate, or in the parabola, to any abscissa and the corresponding ordinate.

(4) In crystallography, the ratio of the three crystallographic axes which determines the position of any plane; the fundamental axial ratio for a given species.

(5) In statistics, a variable entering into the mathematical form of any distribution such that the possible values of the variable correspond to different distributions (any measured quantity of a statistical population that summarizes or describes an aspect of the population).

(6) In computing, a variable that must be given a specific value during the execution of a program or of a procedure within a program.

(7) Limits or boundaries; guidelines; specifications; any constant, definitional or limiting factor (usually in the plural parameters).

(8) Characteristic or a factor; an aspect or element.

(9) In computing syntax for various purposes, an input variable of a function definition, that become an actual value (argument) at execution time (an actual value given to such a formal parameter).

1650-1660: From the French paramètre, from the New Latin parametrum (parameter), the construct being the Ancient Greek παρα- (para-) (beside, subsidiary) + μέτρον (métron) (meter) (measure), from the primitive Indo-European root me- (to measure).  The words was almost exclusive to mathematics & geometry until the late 1920s when it came to be extended to “measurable factor(s) which help to define a particular system", hence the now common alternative meaning “boundary, limit, characteristic factor” (under the influence of perimeter which used a similar spelling and (at least conceptually) could be understood to enjoy some overlap of meaning.  Although the wider definition has been in use since the 1950s, purists have never approved.  Parameter is a noun and parametric & parametrical are adjectives; the noun plural is parameters.

Parameters and perimeters

The more modern ways “parameter” has been used since the early twentieth century does offend the linguistically more fastidious but it seems clear the innovations are here to stay.  Some do however just get it wrong and university lecturers in the social sciences seem to be those who bear the heaviest burden of training a certain number of their institution’s first year students in the correct use of “parameter” & “perimeter”.  That they are sometimes confused is understandable because the spellings are so close and there is some sense of overlap in the meanings, both able to be used in a way which defines limits.  The definitions can be reduced to: (1) perimeter refers to either something physical (a national border; a fence etc) or a representation of something physical (lines on a map; the four sides of a square etc) whereas (2) a parameter is an element of specification, a constant or variable value which can be either an absolute value or a range.  So, a perimeter may be drawn on the basis of certain parameters while the values of parameters will in some cases exist within certain perimeters.  Definitions such as that are vague enough for those so inclined to find contradictions but for the way most people, most of the time (correctly) use parameter & perimeter, it seems serviceable.

Lindsay Lohan and her lawyer in court, Los Angeles, December 2011.