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Monday, September 4, 2023

Photocatalysis

From Wikipedia, the free encyclopedia

 In the experiment above, photons from a light source (out of frame on the right hand side) are absorbed by the surface of the titanium dioxide (TiO
2) disc, exciting electrons within the material. These then react with the water molecules, splitting it into its constituents of hydrogen and oxygen. In this experiment, chemicals dissolved in the water prevent the formation of oxygen, which would otherwise recombine with the hydrogen.

In chemistry, photocatalysis is the acceleration of a photoreaction in the presence of a photocatalyst, the excited state of which "repeatedly interacts with the reaction partners forming reaction intermediates and regenerates itself after each cycle of such interactions." In many cases, the catalyst is a solid that upon irradiation with UV- or visible light generates electron–hole pairs that generate free radicals. Photocatalysts belong to three main groups; heterogeneous, homogeneous, and plasmonic antenna-reactor catalysts. The use of each catalysts depends on the preferred application and required catalysis reaction.

History

Early mentions (1911–1938)

The earliest mention came in 1911, when German chemist Dr. Alexander Eibner integrated the concept in his research of the illumination of zinc oxide (ZnO) on the bleaching of the dark blue pigment, Prussian blue. Around this time, Bruner and Kozak published an article discussing the deterioration of oxalic acid in the presence of uranyl salts under illumination, while in 1913, Landau published an article explaining the phenomenon of photocatalysis. Their contributions led to the development of actinometric measurements, measurements that provide the basis of determining photon flux in photochemical reactions. After a hiatus, in 1921, Baly et al. used ferric hydroxides and colloidal uranium salts as catalysts for the creation of formaldehyde under visible light.

In 1938 Doodeve and Kitchener discovered that TiO
2
, a highly-stable and non-toxic oxide, in the presence of oxygen could act as a photosensitizer for bleaching dyes, as ultraviolet light absorbed by TiO
2
led to the production of active oxygen species on its surface, resulting in the blotching of organic chemicals via photooxidation. This was the first observation of the fundamental characteristics of heterogeneous photocatalysis.

1964–1981

Research in photocatalysis again paused until1964, when V.N. Filimonov investigated isopropanol photooxidation from ZnO and TiO
2
 ; while in 1965 Kato and Mashio, Doerffler and Hauffe, and Ikekawa et al. (1965) explored oxidation/photooxidation of CO
2
and organic solvents from ZnO radiance. In 1970, Formenti et al. and Tanaka and Blyholde observed the oxidation of various alkenes and the photocatalytic decay of N2O, respectively.

A breakthrough occurred in 1972, when Akira Fujishima and Kenichi Honda discovered that electrochemical photolysis of water occurred when a TiO
2
electrode irradiated with ultraviolet light was electrically connected to a platinum electrode. As the ultraviolet light was absorbed by the TiO
2
electrode, electrons flowed from the anode to the platinum cathode where hydrogen gas was produced. This was one of the first instances of hydrogen production from a clean and cost-effective source, as the majority of hydrogen production comes from natural gas reforming and gasification. Fujishima's and Honda's findings led to other advances. In 1977, Nozik discovered that the incorporation of a noble metal in the electrochemical photolysis process, such as platinum and gold, among others, could increase photoactivity, and that an external potential was not required. Wagner and Somorjai (1980) and Sakata and Kawai (1981) delineated hydrogen production on the surface of strontium titanate (SrTiO3) via photogeneration, and the generation of hydrogen and methane from the illumination of TiO
2
and PtO2 in ethanol, respectively.

Photocatalysis has not been developed for commercial purposes. Chu et al. (2017) assessed the future of electrochemical photolysis of water, discussing its major challenge of developing a cost-effective, energy-efficient photoelectrochemical (PEC) tandem cell, which would, “mimic natural photosynthesis".

Types of photocatalysis

Heterogeneous photocatalysis

In heterogeneous catalysis the catalyst is in a different phase from the reactants. Heterogeneous photocatalysis is a discipline which includes a large variety of reactions: mild or total oxidations, dehydrogenation, hydrogen transfer, 18O216O2 and deuterium-alkane isotopic exchange, metal deposition, water detoxification, and gaseous pollutant removal.

Most heterogeneous photocatalysts are transition metal oxides and semiconductors. Unlike metals, which have a continuum of electronic states, semiconductors possess a void energy region where no energy levels are available to promote recombination of an electron and hole produced by photoactivation in the solid. The difference in energy between the filled valence band and the empty conduction band in the MO diagram of a semiconductor is the band gap. When the semiconductor absorbs a photon with energy equal to or greater than the material's band gap, an electron excites from the valence band to the conduction band, generating a electron hole in the valence band. This electron-hole pair is an exciton. The excited electron and hole can recombine and release the energy gained from the excitation of the electron as heat. Such exciton recombination is undesirable and higher levels cost efficiency. Efforts to develop functional photocatalysts often emphasize extending exciton lifetime, improving electron-hole separation using diverse approaches that may rely on structural features such as phase hetero-junctions (e.g. anatase-rutile interfaces), noble-metal nanoparticles, silicon nanowires and substitutional cation doping. The ultimate goal of photocatalyst design is to facilitate reactions of the excited electrons with oxidants to produce reduced products, and/or reactions of the generated holes with reductants to produce oxidized products. Due to the generation of positive holes (h+) and excited electrons (e-), oxidation-reduction reactions take place at the surface of semiconductors irradiated with light.

In one mechanism of the oxidative reaction, holes react with the moisture present on the surface and produce a hydroxyl radical. The reaction starts by photo-induced exciton generation in the metal oxide (MO) surface by photon (hv) absorption:

MO + hν → MO (h+ + e)

Oxidative reactions due to photocatalytic effect:

h+ + H2O → H+ + •OH
2 h+ + 2 H2O → 2 H+ + H2O2
H2O2→ 2 •OH

Reductive reactions due to photocatalytic effect:

e + O2 → •O2
•O2 + HO2• + H+ → H2O2 + O2
H2O2 → 2 •OH

Ultimately, both reactions generate hydroxyl radicals. These radicals are oxidative in nature and nonselective with a redox potential of E0 = +3.06 V. This is significantly greater than many common organic compounds, which typically are not greater than E0 = +2.00 V. This results in the non-selective oxidative behavior of these radicals.

TiO
2
, a wide band-gap semiconductor, is a common choice for heterogeneous catalysis. Inertness to chemical environment and long-term photostability has made TiO
2
an important material in many practical applications. Investigation of TiO2 in the rutile (bandgap 3.0 eV) and anatase (bandgap 3.2 eV) phases is common. The absorption of photons with energy equal to or greater than the band gap of the semiconductor initiates photocatalytic reactions. This produces electron-hole (e /h+) pairs:

Where the electron is in the conduction band and the hole is in the valence band. The irradiated TiO
2
particle can behave as an electron donor or acceptor for molecules in contact with the semiconductor. It can participate in redox reactions with adsorbed species, as the valence band hole is strongly oxidizing while the conduction band electron is strongly reducing.

Homogeneous photocatalysis

In homogeneous photocatalysis, the reactants and the photocatalysts exist in the same phase. The process by which the atmosphere self-cleans and removes large organic compounds is a gas phase homogenous photocatalysis reaction. The ozone process is often referenced when developing many photocatalysts:

Most homogeneous photocatalytic reactions are aqueous phase, with a transition-metal complex photocatalyst. The wide use of transition-metal complexes as photocatalysts is in large part due to the large band gap and high stability of the species. Homogeneous photocatalysts are common in the production of clean hydrogen fuel production, with the notable use of cobalt and iron complexes.

Iron complex hydroxy-radical formation using the ozone process is common in the production of hydrogen fuel (similar to Fenton's reagent process done in low pH conditions without photoexcitation):

Complex-based photocatalysts are semiconductors, and operate under the same electronic properties as heterogeneous catalysts.

Plasmonic antenna-reactor photocatalysis

A plasmonic antenna-reactor photocatalyst is a photocatalyst that combines a catalyst with attached antenna that increases the catalyst's ability to absorb light, thereby increasing its efficiency.

A SiO
2
catalyst combined with an Au light absorber accelerated hydrogen sulfide-to-hydrogen reactions. The process is an alternative to the conventional Claus process that operates at 800–1,000 °C (1,470–1,830 °F).

A Fe catalyst combined with a Cu light absorber can produce hydrogen from ammonia (NH
3
) at ambient temperature using visible light. Conventional Cu-Ru production operates at 650–1,000 °C (1,202–1,832 °F).

Applications

SEM image of wood pulp (dark fibers) and tetrapodal zinc oxide micro particles (white and spiky) in paper.

Photoactive catalysts have been introduced over the last decade, such as TiO
2
and ZnO nano rodes. Most suffer from the fact that they can only perform under UV irradiation due to their band structure. Other photocatalysts, including a graphene-ZnO nanocompound counter this problem.

Paper

Micro-sized ZnO tetrapodal particles added to pilot paper production. The most common are one-dimensional nanostructures, such as nanorods, nanotubes, nanofibers, nanowires, but also nanoplates, nanosheets, nanospheres, tetrapods. ZnO is strongly oxidative, chemically stabile, with enhanced photocatalytic activity, and has a large free-exciton binding energy. It is non-toxic, abundant, biocompatible, biodegradable, environmentally friendly, low cost, and compatible with simple chemical synthesis. ZnO faces limits to its widespread use in photocatalysis under solar radiation. Several approaches have been suggested to overcome this limitation, including doping for reducing the band gap and improving charge carrier separation.

Water splitting

Photocatalytic water splitting separates water into hydrogen and oxygen:

2 H2O → 2 H2 + O2

The most prevalently investigated material, TiO
2
, is inefficient. Mixtures of TiO
2
and nickel oxide (NiO) are more active. NiO allows a significant explоitation of the visible spectrum. One efficient photocatalyst in the UV range is based on sodium tantalite (NaTaO3) doped with lanthanum and loaded with a nickel oxide cocatalyst. The surface is grooved with nanosteps from doping with lanthanum (3–15 nm range, see nanotechnology). The NiO particles are present on the edges, with the oxygen evolving from the grooves.

Self-cleaning glass

Titanium dioxide takes part in self-cleaning glass. Free radicals generated from TiO
2
oxidize organic matter. The rough wedge-like TiO
2
surface can be modified with a hydrophobic monolayer of octadecylphosphonic acid (ODP). TiO
2
surfaces that were plasma etched for 10 seconds and subsequent surface modifications with ODP showed a water contact angle greater than 150◦. The surface was converted into a superhydrophilic surface (water contact angle = 0◦) upon UV illumination, due to rapid decomposition of octadecylphosphonic acid coating resulting from TiO
2
photocatalysis. Due to TiO
2
's wide band gap, light absorption by the semiconductor material and resulting superhydrophilic conversion of undoped TiO
2
requires ultraviolet radiation (wavelength <390 nm) and thereby restricts self-cleaning to outdoor applications.

Disinfection and cleaning

  • Water disinfection/decontamination, a form of solar water disinfection (SODIS). Adsorbents attract organics such as tetrachloroethylene. Adsorbents are placed in packed beds for 18 hours. Spent adsorbents are placed in regeneration fluid, essentially removing organics still attached by passing hot water opposite to the flow of water during adsorption. The regeneration fluid passes through fixed beds of silica gel photocatalysts to remove and decompose remaining organics.
  • TiO
    2
    self-sterilizing coatings (for application to food contact surfaces and in other environments where microbial pathogens spread by indirect contact).
  • Magnetic TiO
    2
    nanoparticle oxidation of organic contaminants agitated using a magnetic field.
  • Sterilization of surgical instruments and removal of fingerprints from electrical and optical components.

Hydrocarbon production from CO
2

TiO
2
conversion of CO
2
into gaseous hydrocarbons. The proposed reaction mechanisms involve the creation of a highly reactive carbon radical from carbon monoxide and carbon dioxide which then reacts with photogenerated protons to ultimately form methane. Efficiencies of TiO
2
-based photocatalysts are low, although nanostructures such as carbon nanotubes and metallic nanoparticles help.

Paints

ePaint is a less-toxic alternative to conventional antifouling marine paints that generates hydrogen peroxide.

Photocatalysis of organic reactions by polypyridyl complexes, porphyrins, or other dyes can produce materials inaccessible by classical approaches. Most photocatalytic dye degradation studies have employed TiO
2
. The anatase form of TiO
2
has higher photon absorption characteristics.

Filtration membranes

Photocatalyst radical generation species allow for the degradation of organic pollutants into non-toxic compounds at a high efficiency. Use of CuO nanosheets to breakdown azo bonds in food dyes is one such example, with 96.99% degradation after only 6 minutes. Degradation of organic matter is a highly applicable property, particularly in waste processing.

The use of photocatalyst TiO2 as a support system for filtration membranes shows promise in improving membrane bioreactors in the treatment of wastewater. Polymer-based membranes have shown reduced fouling and self-cleaning properties in both blended and coated TiO2 membranes. Photocatalyst-coated membranes show the most promise, as the increased surface exposure of the photocatalyst increases its organic degradation activity.

Photocatalysts are also highly effective reducers of toxic heavy metals like hexavalent chromium from water systems. Under visible light the reduction of Cr(VI) by a Ce-ZrO2 sol-gel on a silicon carbide was 97% effective at reducing the heavy metal to trivalent chromium.

Air Filtration

Light2CAT was a project funded by the European Commission from 2012 to 2015. It aimed to develop a modified TiO
2
that can absorb visible light and include this modified TiO
2
into construction concrete. The TiO
2
degrades harmful pollutants such as NOx into NO3. The modified TiO2 is in use in Copenhagen and Holbæk, Denmark, and Valencia, Spain. This “self-cleaning” concrete led to a 5-20% reduction in NOx over the course of a year.

Quantification

ISO 22197-1:2007 specifies a test method for the measurement of NO
2
removal for materials that contain a photocatalyst or have superficial photocatalytic films.

Specific FTIR systems are used to characterize photocatalytic activity or passivity, especially with respect to volatile organic compounds, and representative binder matrices.

Mass spectrometry allows measurement of photocatalytic activity by tracking the decomposition of gaseous pollutants such as nitrogen NOx or CO
2

Pentagram

From Wikipedia, the free encyclopedia
Pentagram

A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around the five points creates a similar symbol referred to as the pentacle, which is used widely by Wiccans and in paganism, or as a sign of life and connections. The word "pentagram" refers only to the five-pointed star, not the surrounding circle of a pentacle.

Pentagrams were used symbolically in ancient Greece and Babylonia. Christians once commonly used the pentagram to represent the five wounds of Jesus.

The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), from πέντε (pente), "five" + γραμμή (grammē), "line". Pentagram refers to just the star and pentacle refers to the star within the circle specifically although there is some overlap in usage. The word pentalpha is a 17th-century revival of a post-classical Greek name of the shape.

History

Early history

Early pentagrams have been found on Sumerian pottery from Ur circa 3500 BCE, and the five-pointed star was at various times the symbol of Ishtar or Marduk.

A Pythagorean "Hugieia Pentagram"
A right-handed interlaced pentagram, popular with Wiccans and some other neo-pagans. The Flag of Morocco often bears the left-handed version.

Pentagram symbols from about 5,000 years ago were found in the Liangzhu culture of China.

The pentagram was known to the ancient Greeks, with a depiction on a vase possibly dating back to the 7th century BCE. Pythagoreanism originated in the 6th century BCE and used the pentagram as a symbol of mutual recognition, of wellbeing, and to recognize good deeds and charity.

From around 300-150 BCE the pentagram stood as the symbol of Jerusalem, marked by the 5 Hebrew letters ירשלם spelling its name.

The word Pentemychos (πεντέμυχος lit. "five corners" or "five recesses") was the title of the cosmogony of Pherecydes of Syros. Here, the "five corners" are where the seeds of Chronos are placed within the Earth in order for the cosmos to appear.

In Neoplatonism, the pentagram was said to have been used as a symbol or sign of recognition by the Pythagoreans, who called the pentagram ὑγιεία hugieia "health".

Western symbolism

Middle Ages

The pentagram was used in ancient times as a Christian symbol for the five senses, or of the five wounds of Christ. The pentagram plays an important symbolic role in the 14th-century English poem Sir Gawain and the Green Knight, in which the symbol decorates the shield of the hero, Gawain. The unnamed poet credits the symbol's origin to King Solomon, and explains that each of the five interconnected points represents a virtue tied to a group of five: Gawain is perfect in his five senses and five fingers, faithful to the Five Wounds of Christ, takes courage from the five joys that Mary had of Jesus, and exemplifies the five virtues of knighthood, which are generosity, friendship, chastity, chivalry, and piety.

The North rose of Amiens Cathedral

The North rose of Amiens Cathedral (built in the 13th century) exhibits a pentagram-based motif. Some sources interpret the unusual downward-pointing star as symbolizing the Holy Spirit descending on people.

Renaissance

Heinrich Cornelius Agrippa and others perpetuated the popularity of the pentagram as a magic symbol, attributing the five neoplatonic elements to the five points, in typical Renaissance fashion.

Romanticism

By the mid-19th century, a further distinction had developed amongst occultists regarding the pentagram's orientation. With a single point upwards it depicted spirit presiding over the four elements of matter, and was essentially "good". However, the influential but controversial writer Éliphas Lévi, known for believing that magic was a real science, had called it evil whenever the symbol appeared the other way up.

  • "A reversed pentagram, with two points projecting upwards, is a symbol of evil and attracts sinister forces because it overturns the proper order of things and demonstrates the triumph of matter over spirit. It is the goat of lust attacking the heavens with its horns, a sign execrated by initiates."
  • "The flaming star, which, when turned upside down, is the heirolgyphic [sic] sign of the goat of black magic, whose head may be drawn in the star, the two horns at the top, the ears to the right and left, the beard at the bottom. It is a sign of antagonism and fatality. It is the goat of lust attacking the heavens with its horns."
  • "Let us keep the figure of the Five-pointed Star always upright, with the topmost triangle pointing to heaven, for it is the seat of wisdom, and if the figure is reversed, perversion and evil will be the result."

The apotropaic (protective) use in German folklore of the pentagram symbol (called Drudenfuss in German) is referred to by Goethe in Faust (1808), where a pentagram prevents Mephistopheles from leaving a room (but did not prevent him from entering by the same way, as the outward pointing corner of the diagram happened to be imperfectly drawn):

Mephistopheles:

I must confess, I'm prevented though
By a little thing that hinders me,
The Druid's-foot on your doorsill–

Faust:

The Pentagram gives you pain?
Then tell me, you Son of Hell,
If that's the case, how did you gain
Entry? Are spirits like you cheated?

Mephistopheles:

Look carefully! It's not completed:
One angle, if you inspect it closely
Has, as you see, been left a little open.

Also protective is the use in Icelandic folklore of a gestured or carved rather than painted pentagram (called smèrhnút in Icelandic), according to 19th century folklorist Jón Árnason:

A butter that comes from the fake vomit is called a fake butter; it looks like something else; but if one makes a sign of a cross over it, or carves a cross on it, or a figure called a buttermilk-knot,* it all explodes into small pieces and becomes like a grain of dross, so that nothing remains of it, except only particles, or it subsides like foam. Therefore it seems more prudent, if a person is offered a horrible butter to eat, or as a fee, to make either mark on it, because a fake butter cannot withstand either a cross mark or a butter-knot.
* The butter-knot is shaped like this: 

East Asian symbolism

Wu Xing's five phases

Wu Xing (Chinese: 五行; pinyin: Wǔ Xíng) are the five phases, or five elements in Taoists Chinese tradition. They are differentiated from the formative ancient Japanese or Greek elements, due to their emphasis on cyclic transformations and change. The five phases are: Fire (火 huǒ), Earth (土 ), Metal (金 jīn), Water (水 shuǐ), and Wood (木 ). The Wuxing is the fundamental philosophy and doctrine of traditional Chinese Medicine and Acupuncture.

Uses in modern occultism

Based on Renaissance-era occultism, the pentagram found its way into the symbolism of modern occultists. Its major use is a continuation of the ancient Babylonian use of the pentagram as an apotropaic charm to protect against evil forces. Éliphas Lévi claimed that "The Pentagram expresses the mind's domination over the elements and it is by this sign that we bind the demons of the air, the spirits of fire, the spectres of water, and the ghosts of earth." In this spirit, the Hermetic Order of the Golden Dawn developed the use of the pentagram in the lesser banishing ritual of the pentagram, which is still used to this day by those who practice Golden Dawn-type magic.

Aleister Crowley made use of the pentagram in his Thelemic system of magick: an adverse or inverted pentagram represents the descent of spirit into matter, according to the interpretation of Lon Milo DuQuette. Crowley contradicted his old comrades in the Hermetic Order of the Golden Dawn, who, following Levi, considered this orientation of the symbol evil and associated it with the triumph of matter over spirit.

Use in new religious movements

Baháʼí Faith

Haykal by the Báb written in his own hand

The five-pointed star is a symbol of the Baháʼí Faith. In the Baháʼí Faith, the star is known as the Haykal (Arabic: "temple"), and it was initiated and established by the Báb. The Báb and Bahá'u'lláh wrote various works in the form of a pentagram.

The Church of Jesus Christ of Latter-day Saints

The Church of Jesus Christ of Latter-day Saints is theorized to have begun using both upright and inverted five-pointed stars in Temple architecture, dating from the Nauvoo Illinois Temple dedicated on 30 April 1846. Other temples decorated with five-pointed stars in both orientations include the Salt Lake Temple and the Logan Utah Temple. These usages come from the symbolism found in Revelation chapter 12: "And there appeared a great wonder in heaven; a woman clothed with the sun, and the moon under her feet, and upon her head a crown of twelve stars."

Wicca

Typical Neopagan pentagram (circumscribed)
 
USVA headstone emblem 37

Because of a perceived association with Satanism and occultism, many United States schools in the late 1990s sought to prevent students from displaying the pentagram on clothing or jewelry. In public schools, such actions by administrators were determined in 2000 to be in violation of students' First Amendment right to free exercise of religion.

The encircled pentagram (referred to as a pentacle by the plaintiffs) was added to the list of 38 approved religious symbols to be placed on the tombstones of fallen service members at Arlington National Cemetery on 24 April 2007. The decision was made following ten applications from families of fallen soldiers who practiced Wicca. The government paid the families US$225,000 to settle their pending lawsuits.

Other religious use

Satanism

The inverted pentagram is the most notable and widespread symbol of Satanism.
 
The Sigil of Baphomet, the official insignia of the Church of Satan and LaVeyan Satanism

The inverted pentagram is the symbol used for Satanism, sometimes depicted with the goat's head of Baphomet within it, which originated from the Church of Satan. In some depictions the devil is depicted, like Baphomet, as a goat, therefore the goat and goat's head is a significant symbol throughout Satanism. The inverted pentagram is also used as the logo for The Satanic Temple, which also featured a depiction of Baphomet's head. The Sigil of Baphomet is also adopted by the Joy of Satan Ministries who instead incorporate cuneiform script, attributing it to the earliest use of the pentagram in Sumeria.

Serer religion

The five-pointed star is a symbol of the Serer religion and the Serer people of West Africa. Called Yoonir in their language, it symbolizes the universe in the Serer creation myth, and also represents the star Sirius.

Judaism

The pentagram has been used in Judaism since at least 300BCE when it first was used as the stamp of Jerusalem. It is used to represent justice, mercy, and wisdom.

Other modern use

  • The pentagram is featured on the national flags of Morocco (adopted 1915) and Ethiopia (adopted 1996 and readopted 2009)
  • The Order of the Eastern Star, an organization (established 1850) associated with Freemasonry, uses a pentagram as its symbol, with the five isosceles triangles of the points colored blue, yellow, white, green, and red. In most Grand Chapters the pentagram is used pointing down, but in a few, it is pointing up. Grand Chapter officers often have a pentagon inscribed around the star (the emblem shown here is from the Prince Hall Association).
  • A pentagram is featured on the flag of the Dutch city of Haaksbergen, as well on its coat of arms.
  • A pentagram is featured on the flag of the Japanese city of Nagasaki, as well on its emblem.

Geometry

Koch snowflakes drawn with MSWLogo (in Tartapelago)

The pentagram is the simplest regular star polygon. The pentagram contains ten points (the five points of the star, and the five vertices of the inner pentagon) and fifteen line segments. It is represented by the Schläfli symbol {5/2}. Like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as its symmetry group the dihedral group of order 10.

It can be seen as a net of a pentagonal pyramid although with isosceles triangles.

Construction

The pentagram can be constructed by connecting alternate vertices of a pentagon; see details of the construction. It can also be constructed as a stellation of a pentagon, by extending the edges of a pentagon until the lines intersect.

Golden ratio

A regular pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

The golden ratio, φ = (1 + 5) / 2 ≈ 1.618, satisfying

plays an important role in regular pentagons and pentagrams. Each intersection of edges sections the edges in the golden ratio: the ratio of the length of the edge to the longer segment is φ, as is the length of the longer segment to the shorter. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is φ. As the four-color illustration shows:

The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon.

Trigonometric values

As a result, in an isosceles triangle with one or two angles of 36°, the longer of the two side lengths is φ times that of the shorter of the two, both in the case of the acute as in the case of the obtuse triangle.

Spherical pentagram

A pentagram can be drawn as a star polygon on a sphere, composed of five great circle arcs, whose all internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio (Description of the wonderful rule of logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). It was studied later by Carl Friedrich Gauss.

Three-dimensional figures

Several polyhedra incorporate pentagrams:

Higher dimensions

Orthogonal projections of higher dimensional polytopes can also create pentagrammic figures:

4D 5D

The regular 5-cell (4-simplex) has five vertices and 10 edges.

The rectified 5-cell has 10 vertices and 30 edges.

The rectified 5-simplex has 15 vertices, seen in this orthogonal projection as three nested pentagrams.

The birectified 5-simplex has 20 vertices, seen in this orthogonal projection as four overlapping pentagrams.

All ten 4-dimensional Schläfli–Hess 4-polytopes have either pentagrammic faces or vertex figure elements.

Pentagram of Venus

The pentagram of Venus

The pentagram of Venus is the apparent path of the planet Venus as observed from Earth. Successive inferior conjunctions of Venus repeat with an orbital resonance of approximately 13:8—that is, Venus orbits the Sun approximately 13 times for every eight orbits of Earth—shifting 144° at each inferior conjunction. The tips of the five loops at the center of the figure have the same geometric relationship to one another as the five vertices, or points, of a pentagram, and each group of five intersections equidistant from the figure's center have the same geometric relationship.

In computer systems

The pentagram has these Unicode code points that enable them to be included in documents:

  • U+26E4 PENTAGRAM
  • U+26E5 RIGHT-HANDED INTERLACED PENTAGRAM
  • U+26E6 LEFT-HANDED INTERLACED PENTAGRAM
  • U+26E7 INVERTED PENTAGRAM

Inequality (mathematics)

From Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Inequality...