Resonance (Chemistry)
Introduction (all levels)
Abbreviations: e- = electron
In chemistry we make simple drawings showing the location of each atom in a molecule.
We call these Lewis diagrams or dot diagrams.
Examples:
Here is a slightly more complicated Lewis diagram. This is Tyrosine. Don’t worry about what the molecule is; the point is only that we draw it with simple shapes.
Single lines represent single-bonds between atoms
Double-lines represent double-bonds between atoms
We use Lewis diagrams all the time.
Yet once in a while the rules for making Lewis structures don’t work
Here are three ways that the rules break down, even for a simple molecule:
1. A molecule might have 2 or 3 different ways of being drawn. How do we decide which way is correct? The rules don’t tell us.
2. Using the same example (ozone) the rules seem to want us to draw it as a straight line. Yet experimental measurements show that ozone is bent.
Hmm, the rules fail to predict if a molecule is straight or bent.
electron clouds of ozone molecule
3. The rules sometimes compel us to make double bonds on side of a molecule, and single bonds on the other side.
Double bonds are stronger, holding atoms closer together, so they are shorter length.
Single bonds are weaker, holding atoms less close together, so they are longer length.
Yet sometimes experimental measurements of a molecule show that all bonds are the same length.
Consider the possible Lewis structures for ozone: Either way, some bonds are shorter while others are longer. Yet the 3D electron map for ozone (above) shows that all bond lengths are identical.
The rules, clearly, are incorrect: Neither model is correct enough.
Takeaway: Even for small, simple molecules, Lewis structure rules can fail in multiple ways.
This is a problem yet also an opportunity: In physics, when a rule break down, that means the “rule” is really a simplified case of a more general rule.
Imperfect assumptions -> imperfect rules
What assumptions were behind Lewis structure rules?
Atoms assumed to be like tiny solar system.
protons, neutrons, and e- are assumed to be in in one location or in another. location.
Yet experiments prove that e- aren’t like tiny billiard balls at all.
They aren’t solid objects!
We’ll need some quantum mechanics to figure out what e- really are, but for now we can say this:
e- act like a cloud of energy that spreads out.
Resonance: Fudging drawing rules to make them fit real world measurements
When more than one possible Lewis structure for a molecule can be drawn, resonance is a trick in which we draw all possible forms, and say that the real molecule is an average of each separate drawing.
We call this a resonance hybrid model.
Several Lewis structures are collectively describe the true structure.
Example: Benzene is a hydrocarbon, C6H6. The Lewis rules are ambiguous. We could draw it either way. Which structure is right? Neither!
We just draw both possibilities, and say that the real molecule is an average of the two possibilities.
The real structure is an average of the two different drawings. (See the top 2 drawings here.)
The bottom picture is a way to symbolize that some of these e- aren’t in the left form, or the right form, but are really sort of evenly spread out.
Now combine this idea with what we learned previously in chemistry:
There are many different ways to draw the same molecule.
All of the following are different ways of showing benzene.
Honors: ozone
According to experimental evidence from microwave spectroscopy, ozone is a bent molecule.
The central atom is sp² hybridized with one lone pair.
It can be represented as a resonance hybrid with two contributing structures, each with a single bond on one side and double bond on the other.
Here’s one way to draw the resonance structures of ozone.
Here is a better way of drawing it, with the experimentally determined angles.
External links
TBA
Learning Standards
NGSS
HS-PS1-1. Use the periodic table as a model to predict the relative properties of elements based on the patterns of electrons in the outermost energy level of atoms.
NGSS Evidence Statement – Students predict the following patterns of properties:
i. The number and types of bonds formed (i.e. ionic, covalent, metallic) by an element and
between elements;
College Board Standards for College Success: Science
Objective C.1.2 Electrons
Students understand that the interactions of electrons between and within atoms are the primary factors that determine the properties of matter.
ESSENTIAL KNOWLEDGE – Students apply, as well as engage and reason with, the following concepts in the performance expectations:
• Atoms can bond to form molecules, ionic lattices, network covalent structures or materials with metallic properties. Each of these types of structures has different, yet predictable, properties that depend on the identity of the elements and the types of bonds formed.
• The forces of attraction between the particles in molecules, ionic lattices, network covalent structures or materials with metallic properties are called chemical bonds.
• The bonds in most compounds fall on a continuum between the two extreme models of bonding: ionic and covalent.
• An ionic bond involves the attraction between two oppositely charged ions, typically a positively charged metal ion and a negatively charged nonmetal ion. An ion attracts oppositely charged ions from every direction, resulting in the formation of three-dimensional lattices.
• Covalent bonds typically involve at least two electrons shared between the bonding atoms. Nonmetal atoms usually combine by forming one or more covalent bonds between atoms. Covalent bonding can result in the formation of structures ranging from
small molecules to large molar mass biopolymers and three-dimensional lattices (e.g., a diamond).
Objective C.1.4 Representations of Matter
Students understand that atoms, molecules and ionic substances can be represented with a variety of models.
C-PE.1.4.1 Translate among representations (including molecular formulas, Lewis structures, ball-and-stick models and space-filling models) of macroscopic, atomic–molecular and symbolic levels of matter. Compare and contrast the types of information that can be inferred from the different representations. Choose the most appropriate representation to illustrate a physical or chemical process
C-PE.1.4.2 Construct Lewis structures for simple molecules, showing all bonds and lone pairs of electrons for simple molecules. Using regions of electron density, predict electron pair geometry and the shape of the molecule from the arrangement of the atoms in space.
Flint Michigan Water Crisis
The Flint water crisis was a public health crisis from 2014 to 2019 in Flint, Michigan.
In April 2014, Flint changed its water source from treated Detroit Water and Sewerage Department water (sourced from Lake Huron and the Detroit River) to the Flint River.
Officials failed to apply corrosion inhibitors to the water.
As a result, lead from aging lead pipes leached into the water supply.
This led to extremely elevated levels of the heavy metal lead, exposing over 100,000 residents to elevated lead levels.
Click; Scroll down to “Turning on the tap in America”
Then click on the city of Flint, MI. Go through that presentation.
Five-month-old Dakota Erler of Flint gets blood drawn from her heel in order to have her lead levels tested at Carriage Town Ministries in 2016.
5-month-old Dakota Erler of Flint, Mich., gets blood drawn from her heel in order to have her blood lead levels tested at Carriage Town Ministries in Flint, Mich., on Thursday, February 4, 2016. Erler’s father, Don Erler, said she has been drinking bottled water sine birth however they wanted to be sure of any exposure that could occur from bathing. Several blood lead level testing events have been put on in partnership with the Michigan State Health Department following the declaration for a state of emergency in the city of Flint due to the ongoing water crisis. (Photo by Brittany Greeson for The Washington Post)
This is how the process was supposed to have worked.
The water supply company adds a corrosion inhibitor (orthophosphate) to the water supply.
This creates a coating on city pipes as well as pipes the customer’s property.
This prevents the pipes from leaching lead.
This is how the process actually worked in Flint Michigan.
Without corrosion control.
What did the water look like?
Association between water discoloration and lead concentrations for water samples collected in January 2015.
A photo of the original sample bottles is overlaid with a bar chart of lead in water concentrations.
This next image is from Flint Water Crisis Caused By Interrupted Corrosion Control: Investigating “Ground Zero” Home
Videos
Chemistry and the Flint Water Crisis (YouTube)
Details of the chemistry
This next image is from Flint Water Crisis Caused By Interrupted Corrosion Control: Investigating “Ground Zero” Home
and
Chemical & Engineering News
F
G
Plotting Coulomb’s law or the law of gravity – not quite hyperbolas
Here’s a graph of force versus distance using an inverse square law.
This is Coulomb’s law, showing the magnitude of the force between two electrically charged particles.
It looks hyperbolic – but does this actually qualify as a hyperbola?
What is a hyperbola?
There are many different yet equivalent definitions for hyperbolas, see those definitions here:
Hyperbola, Math Is Fun, The Hyperbola, Lumen, Graphs of Hyperbolas Centered at the Origin, CK-12
For our graph:
Force is plotted on the Y-axis.
‘r’ is the distance between two charged objects, plotted on the X-axis.
In the above example we used Coulomb’s law, but mathematically it is the same form as Newton’s law of universal gravitation:
K is just a constant. With gravity this constant is extremely small.
With electric attraction/repulsion the constant is many orders of magnitude larger.
So for any of these cases, is this curve a hyperbola?
No. Hyperbolas – by definition – are conic sections.
And by definition conic sections must be able to be put into this format:
Ax2 + Bxy + Cy2 +Dx + Ey + F = 0
The above equations – Coulomb’s law and Newton’s law – can’t be put into this format. Thus these curves cannot be hyperbolic.
Rational functions
So what kind of curve are these force vs distance curves?
They are not hyperbolas but they are rational functions: the ratio of two polynomials.
It is called “rational” because one is divided by the other, like a ratio.
Notice that rational functions have horizontal and vertical asymptotes, and inverse relationships, so they visually approximate hyperbolas.
One might even say that they share some properties of hyperbolas without fulfilling all the criteria of actually being one.
A special case of rational functions
Although not applicable for Coulomb’s law, one may note that rational functions of the form (ax+b)/(cx+d) are hyperbolas
As long as determinant, ad-bc, and c, are non-zero.
So hyperbolas are special cases of rational functions.
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Thanks for visiting. While you are here see our other articles on mathematics.
#hyperbolas #conic #rationalequations
The Eötvös effect
The Eötvös effect is the change in perceived gravitational force caused by the change in centrifugal acceleration resulting from eastbound or westbound velocity.
The measured effect is caused by the motion of the object traveling with, or against, the rotation of the Earth.
When moving eastbound, the object’s angular velocity is increased (in addition to Earth’s rotation)
thus the centrifugal force also increases, causing a perceived reduction in gravitational force.
When moving westbound, the object’s angular velocity is decreased,
thus the centrifugal force decreases, causing a perceived increase in gravitational force.
From flatearth.ws, debunking flat earth misconceptions
In the early 1900s (decade), a German team from the Institute of Geodesy in Potsdam carried out gravity measurements on moving ships in the Atlantic, Indian, and Pacific oceans.
While studying their results, the Hungarian nobleman and physicist Baron Roland von Eötvös (Loránd Eötvös) noticed that the readings were lower when the boat moved eastwards, higher when it moved westward. He identified this as primarily a consequence of Earth’s rotation.
In 1908, new measurements were made in the Black Sea on two ships, one moving eastward and one westward. The results substantiated Eötvös’ understanding.
Relationship between eötvös effect and Coriolis effect
Some people say that the Eötvös effect is the vertical component of the Coriolis effect. Max on Physics StackExchange explains to us
In many science disciplines, casual versus formal usages become intermixed, and this is certainly one area.
Eötvös is not the vertical component of Coriolis.
The earth is both (a) spherical and (b) spinning. This produces a number of phenomena that affect bodies in motion on or near the surface of the Earth.
In casual usage these phenomena tend to be lumped together into all being called “Coriolis,” but they are actually discrete physical properties that are not related, except for the fact that they are artifacts of (a), (b), or both.
Coriolis is a conservation of angular momentum consideration when objects move north/south across a spinning sphere.
As you move away from the equator latitudinally, the same angular rate of rotation around the Earth’s C/G results in a different velocity in the east/west component, and the effects of this difference is the Coriolis Effect.
Were the Earth a cylinder instead of a sphere, there’d be no Coriolis Force. (*)
Eötvös on the other hand is a centrifugal force/orbital mechanics problem. Eötvös would still occur on a cylinder, where Coriolis would not.
There is an angular momentum force that acts east/west based on the height of an object’s trajectory or orbit, and thus would affect the vertical component of a projectile’s trajectory at long distances involving high trajectories.
But this isn’t Eötvös at all. If I shoot a projectile perfectly vertically a few miles into the air, conservation of angular momentum dictates the projectile will not land back on me, it will land several feet west of me, opposite the direction of the Earth’s spin. It may be more correct to think of this motion as the vertical component of Coriolis.
(*) This gets addressed later on this page. There would be some force, but it would different from what we see on a spherical Earth.
Navigation – Science, History, and Cultural connections
How did humans learn how to navigate the world? No one person, from any one culture, ever figured it all out on their own. Navigation developed over time, over the globe, with contributions from different people and cultures. This is a story of world history, of science, of astronomy, of story telling, and of multicultural collaboration over the millennia.
Earliest known maritime navigation
The Austronesian peoples are a large group of various peoples from Taiwan, maritime Southeast Asia, Oceania and Madagascar, united by a language family.
They appear to originate from a prehistoric seaborne migration from Taiwan, somewhere between 5000 to 1500 BCE.
Peoples from these migrations reached parts of the Philippines around 4000 BCE.
They were the first people to invent maritime sailing technology – catamarans, outrigger boats, lashed-lug boat building, and the crab claw sail.
Around 1500 BCE some groups from the Philippines went out and colonized the islands of Micronesia (thousands of small islands in the western Pacific Ocean.)
By about 900 BCE their descendants had spread across the Pacific, reaching Tonga and Samoa. In this region, a distinctive Polynesian culture developed.
Polynesian navigators used a range of tools and methods:
observation of birds, star navigation, and use of waves and swells to detect nearby land.
Songs, mythological stories, and star charts were used to remember navigational information.
(text in this section excerpted and adapted from Wikipedia. Austronesian peoples, Polynesian navigation)
See the PBS TV show Polynesia’s Genius Navigators
Here is the hand method used to find the altitude of the Polaris, to estimate one’s latitude.
from Ben Finney et al., Re-learning a vanishing art
We read in the article (Re-learning a vanishing art, Ben R. Finney)
The clearing skies after the storm enabled Nainoa to make multiple star observations in order to determine latitude. For example, on the night of March 23-24 he was able to estimate that the canoe should reach 10°N around sunrise by measuring, with his outstretched hand, the angular height of Polaris (Figure 7) and the angular height of Acrux (the bottom star of the Southern Cross when upright, Figure 8).
In addition, observations of two other stars confirmed the latitude. This measurement turned out to be only 21 miles off, for at sunrise the canoe was actually 21 minutes of latitude north of 10 °N.
We see this same method used in the Disney film, Moana
“Waialiki using this technique to measure the altitude of a group of stars. Look closely and you can see that she’s measuring the stars in Orion’s Belt. The position of Moana’s hand indicates the star above her index finger has an altitude of 21º. Given that the movie takes place about 2,000 years ago near Samoa, the position of Orion indicates they are travelling exactly due East.”
Screen shot from Disney movie Moana
How far they’ll go: Moana shows the power of Polynesian celestial navigation, Duane W. Hamacher and Carla Bento Guedes, The Conversation
Rebbelib aka Marshall Islands stick chart
Made of bamboo and shell maps.
Possibly used since circa 1000 CE, right up to the modern era.
This comparison of a stick chart with a more modern map was made by Reddit user lander ceuppens.
These charts show
major ocean swell patterns; the ways that islands disrupt those patterns,
made from coconut fronds and shells
Islands are represented by shells tied to the framework, or by lashed junction of 2 or more sticks.
Threads represent prevailing ocean surface wave-crests and directions they took as they approached islands
Charts varied so much in form and interpretation that the individual navigator who made the chart was the only person who could fully interpret and use it.
Use of these ended after World War II when new technologies made navigation more accessible.
Navigation of the ancient Mediterranean
We read from By Dr. Markus Nielbock
Around the middle of the 4th millennium BCE, Egyptian ships sailed the eastern Mediterranean and established trade routes with Byblos in Phoenicia, the biblical Canaan, now Lebanon. This is about the time when the Bronze Age began.”
… Soon, the navigators realised that celestial objects, especially stars, can be used to keep the course of a ship. Such skills have been mentioned in early literature like Homer’s Odyssey which is believed to date back to 8th century BCE.
The original sources are thought to originate from the Bronze Age, in which the Minoans of Crete were a particularly influential people. They lived between 3,650 and 1,450 BCE.
Navigation in the Ancient Mediterranean and Beyond, By Dr. Markus Nielbock
Navigation in the Ancient Mediterranean and Beyond, Astro.edu
Minoan, Greek, and the Phoenician civilizations of the ancient Mediterranean.
Around 200 BCE the Antikythera mechanism was invented. Shockingly, this was a simple analogue computer and orrery. It was used to predict astronomical positions and eclipses for calendrical and astrological purposes. It and its possible sister devices may have been used in navigation.
800 – 1100 CE Viking Navigators
PBS Nova Secrets of Viking Ships
and PBS The Vikings
1200s-1400s Swahili East African coast
The Swahili coast is a coastal area of Southeast Africa on the Indian Ocean. It is inhabited by the Swahili people. It includes the coastal regions of Mozambique, Kenya, and Tanzania as well as the coastal islands.
This area was historically known as Azania or Zingion in the Greco-Roman era, and as Zanj or Zinj in Middle Eastern, Chinese and Indian literature from the 7th to the 14th century.
coastal elites invested in long‐distance voyages at least as early as the 13th century and that a small number of ships owned by Swahili patricians sailed as far as Arabia and, from the 16th to mid–18th centuries, Western India (Vernet in press).”
… Yemeni sources confirm that, during that century, ships from Mogadishu made annual trips to Aden, al‐Shihr, and other Hadramawt ports… the general prosperity of the years 1200–1340 led the Swahili to increase regional navigation and develop regular overseas trade.
When Did the Swahili Become Maritime? Jeffrey Fleisher et al., American Anthropologist, Vol 117, No 1, 3/2015
1300s CE Jacob’s Staff
In 1342 CE the Jewish scholar Levi ben Gershon, known as Gersonides, wrote “On Sines, Chords and Arcs,” proving the sine law for plane triangles and giving five-figure sine tables.
Gersonides also created the astronomical, surveying, and navigational device known as the Jacob’s Staff.
1400s CE Ancient Chinese Explorers
Ancient Chinese Explorers
A century before Europeans ‘discovered’ the Indian Ocean, Chinese merchants led by the redoubtable Zheng He (1371-1433) journeyed as far as Zanzibar. See the PBS NOVA program Sultan’s Lost Treasure (2001)
1400s CE Astrolabe
Abraham Zacuto (Abraão ben Samuel Zacuto) 1452 – 1515 CE
A Spanish Jewish astronomer, mathematician, rabbi and historian. Served as Royal Astronomer to King John II of Portugal.
He developed a new type of astrolabe, specialized for practical determination of latitude while at sea, in contrast to earlier multipurpose devices intended for use ashore.
(I couldn’t find a good picture of original version. This photo is from a more advanced model in the 1800s.)
18th century astrolabe made in North Africa.. Photo by Evan Bench, https://www.flickr.com/photos/austinevan/3317031220
1400s CE Navigational almanac
Abraham Zacuto’s principal claim to fame is the great astronomical treatise, written while he was in Salamanca, begun around 1470 and completed in 1478.
It was composed of detailed astronomical tables (ephemerides), with radix set in year 1473 and the meridian at Salamanca, charting the positions of the Sun, Moon and five planets. The calculations were based on the Alfonsine Tables and the works of earlier astronomers (notably of the 14th-century Majorcan school
He set the data in a simple format, with the positions of a planet easily interpolated between entries, making it quite easy to use.
His disciple Joseph Vizinus adapted it into a Latin translation. This revolutionized ocean navigation. Prior to the Almanac, navigators seeking to determine their position had to correct for compass error (deviation of the magnetic north from the true north) by recourse to the quadrant and the Pole Star. But this proved less useful as they approached the equator and the Pole Star began to disappear into the horizon.
Zacuto’s Almanach supplied the first accurate table of solar declination, allowing navigators to use the sun instead. As the quadrant could not be used to look directly at the sun, Portuguese navigators began using the astrolabe on board (an old land-based instrument to measure the height of the sun indirectly).
Zacuto’s tables in conjunction with the new metal nautical astrolabe allowed navigators to take accurate readings anywhere. Already in 1497, Vasco da Gama took Zacuto’s tables and the astrolabe with him on the maiden trip to India.
The above section excerpted and adapted from Abraham Zacuto, Wikipedia.
Z is for Abraham Zacuto, Zacuto, Biographical Encyclopedia of Astronomers
1500s CE Mercator map
Perhaps the most useful map made for ocean navigation. Read all about it here, Mercator maps: Use and criticism
1700’s-1800s CE The Binnacle
One of the great advances in navigational equipment was the binnacle: This is a waist-high case found on the deck of a ship, that holds the compass. It is mounted in gimbals to keep it level while the ship pitched and rolled. It has a mechanism to compensate for errors in detecting the Earth’s magnetic field. Every ship’s captain would use one.
With the introduction of iron-clad ships the magnetic deviation observed in compasses became more severe. Methods of compensation by arranging iron or magnetic objects near the binnacle were developed.
Lord Kelvin patented in the 1880s a system of compass and which incorporated two compensating spheres. These are colloquially known as “Kelvin’s balls” in the UK, and “navigator’s balls” in the United States
The Binnacle
Photo by RK
The Chronometer
Making the sea clock practical was critical to improving sea navigation. It was the only way to keep track of the ship’s latitude. See the Chronometer
See the PBS show “Lost at Sea: The Search for Longitude,” and the book Longitude by Dava Sobel.
The Search for Longitude
The Sextant
A sextant is a doubly reflecting navigation instrument that measures the angular distance between two visible objects.
It measures the angle between an astronomical object and the horizon for the purposes of celestial navigation.
Sextant by E. & G. W. Blunt, New York, and Quadrant by David White Co. for the US Navy, c. 1930 – Museum of Science and Industry (Chicago) Wikimedia Commons
Justin Shin points out that we can turn ourselves into (approximate) human sextants!
Point at the horizon with one arm and at the north star with the other arm. The angle that your arms make gives you your latitude on the planet.
Pointing to the horizon gives you a tangent line to a circular earth. Pointing at the north star give you a line parallel to the earth’s north/south axis, Everything else follows pretty neatly
Resources
PBS NOVA Secrets of Ancient Navigators
Celestial navigation in the Classroom
Re-learning a vanishing art, Ben R. Finney et al., p.41-90, The Journal of the Polynesian Society, Vol. 95, No 1, 1986
Learning Standards
Ocean Literacy Scope and Sequence for Grades K-12
b) The ocean provides food, medicines, and mineral and energy resources. It supports jobs and national economies, serves as a highway for transportation of goods and people, and plays a role in national security.
c) The ocean is a source of inspiration, recreation, rejuvenation, and discovery. It is also an important element in the heritage of many cultures.
National Standards for History Basic Edition, 1996
5-12 Identify major technological developments in shipbuilding, navigation, and naval warfare and trace the cultural origins of various innovations.
Massachusetts History and Social Science Curriculum Framework
The Political, Intellectual and Economic Growth of the Colonies. Explain the importance of maritime commerce in the development of the economy of colonial Massachusetts, using historical societies and museums as needed.
A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)
Some objects in the solar system can be seen with the naked eye. Planets in the night sky change positions and are not always visible from Earth as they orbit the sun. Stars appear in patterns called constellations, which can be used for navigation and appear to move together across the sky because of Earth’s rotation…. The solar system consists of the sun and a collection of objects, including planets, their moons, and asteroids that are held in orbit around the sun by its gravitational pull on them. This model of the solar system can explain tides, eclipses of the sun and the moon, and the motion of the planets in the sky relative to the stars.
Cultural and historical origin of trigonometry
Cultural and historical origin of trigonometry
Hipparchus of Nicaea 190 – 120 BCE was a Hellenic (“Greek”) astronomer, geographer, and mathematician, considered the founder of trigonometry. He was born in Nicaea, Bithynia – now İznik, Turkey. Ethnically he was a Bithynian.
Hipparchus effectively developed the predecessor of Sine and Cosine through his study of chords, straight line segments whose endpoints both lie on a circular arc. He developed this for a practical reason – he was computing the eccentricity of the orbits of the Moon and Sun.
This work was continued by Claudius Ptolemy (Κλαύδιος Πτολεμαῖος) of Roman Egypt (90–168 CE). He was a Macedonian Hellenist mathematician, astronomer, and geographer.
He was a Macedonian.
Hypatia ~360-415 CE. She was a Hellenistic Neoplatonist philosopher, astronomer, and mathematician, who lived in Alexandria, Egypt, then part of the Eastern Roman Empire.
“Hypatia was known more for the work she did in mathematics than in astronomy, primarily for her work on the ideas of conic sections introduced by Apollonius. She edited the work On the Conics of Apollonius, which divided cones into different parts by a plane. This concept developed the ideas of hyperbolas, parabolas, and ellipses.
With Hypatia’s work on this important book, she made the concepts easier to understand, thus making the work survive through many centuries. Hypatia was the first woman to have such a profound impact on the survival of early thought in mathematics.”
Hypatia, Biographies of women mathematicians
There was much trade and scholarly contact between ancient Greece and India after the Indian campaign of Alexander the Great.
We soon see the continuation of trigonometry in India.
Wikimedia, Roman trade in the subcontinent according to the Periplus Maris Erythraei 1st century CE
Some classic, 4th to 5th century CE Siddhantas defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine.
They used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine.
The words jya and kojya eventually became sine and cosine respectively after a mistranslation described above.
The classic Sanskrit astronomy text, the Sūrya Siddhānta, thought to have been composed around 800 CE. It provides table of sines function which parallel the Hipparchian table of chords, though the Indian calculations are more accurate.
This Indian trigonometry was copied and translated by Arabic scholars, where it soon had a great effect in stimulating the Arabic sciences.
In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer in spherical trigonometry.
Muhammad ibn Jābir al-Harrānī al-Battānī (known as Al-Battani, or Albatenius) (853-929 AD) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants.
By the 10th century AD, in the work of Abū al-Wafā’ al-Būzjānī, Muslim mathematicians were using all six trigonometric functions.
In 1342, the Jewish scholar Levi ben Gershon, known as Gersonides, wrote “On Sines, Chords and Arcs,” proving the sine law for plane triangles and giving five-figure sine tables.
Gersonides also created the astronomical, surveying, and navigational device known as the Jacob’s Staff.
from Peter Apian’s Cosmographia, 1524 CE
#mathematic #history
Harden servers and computers
How does one harden a computer/server?
Keep Your Server’s Operating Systems Updated
Enforce The Use of Strong Passwords
Have a robust account lockout policy. Lock accounts after small number of incorrect attempts.
Update or Remove Third Party Software
Remove any software not required for normal server operation.
CIS Benchmarks for hardening – Suggested settings Windows Security Policy, User Rights Assignments, Audit and Event Log Policy. [Center for Internet Security]
Turn off all of the automatic accounts
Shut down the ports and processes not needed for operation
Radians
Learning goals: Students will understand
why we traditionally measure angles in degrees
that the usually stated number of degrees in a circle is a social/cultural convention, not a scientific or mathematical fact
that there exists a purely natural measure of angles (radians)
that natural units are not social/cultural conventions: rather they are relationships that exist independently of human choice.
We usually measure angles in degrees. But today we’re going to learn about another way to measure angles, called radians.
First let’s think about everyday life: Outside of school, in any construction project, we measure angles in degrees.
That just seems like the only and obvious way.
Plumbing and Heating (46.0503) T-Chart
We’re told that there are 360 degrees in a circle.
But once we get to high school trigonometry we start using a different way of measuring angles.
Asking “Why change?” might be wrong question.
A better question would be “Why did we start measuring angles in degrees to begin with?”
Because 2500 years ago ancient Sumerians discovered that there were about 360 days in a year, that’s why.
This knowledge was picked up by the ancient Babylonians.
That’s not exactly correct – we now know that there are about 365.25 days in a year.
Geography connections: Where was ancient Babylon?
Because of this, the ancient Sumerians and Babylonians used 360 and 60 as building blocks for their counting systems.
The Babylonians knew, of course, that the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle, in fact that was evidently the reason why they chose to divide the circle into 360 degrees (and we are still burdened with that figure to this day).
A History of Pi, Petr Beckmann
That system then passed down to later Greek and Roman civilizations, Islamic civilizations, and then to the European and Asian civilizations.
We’re so used to using it that dividing a circle into 360 parts seems natural. But there’s nothing necessary or natural about this. We didn’t discover this – we defined this. It’s just a convention.
The degree is an arbitrary unit; basically any division of a circle would work as a system of measurement. The degree has the advantage that 360 divides evenly by 2, 3, 4, 5, 6, 8, 9 & 10 making it easy to mentally calculate an angle; indeed this is the major advantage of all old imperial units.
Ray Gallagher, Belfast, Northern Ireland
We could have chosen any other number. During the French Revolution there was an attempt to make a metric system version of angle measurement.
Metric right angles were defined as having 100 “degrees”
(obviously, these degrees were smaller than regular degrees.)
A metic circle would had 400 metric degrees instead of 360.
To avoid confusion these slightly smaller degrees were called gradians, or the gon grad.
360 degrees = 400 gradians
This system was used by some surveyors and engineers in Europe for some time. It is not common anymore. Here we see a French compass from 1922, divided into 400 degrees.
compassmuseum.com/diverstext/divisions.htm
From Compassipedia, part 2, the division systems.
In the end this new metric degree system didn’t catch on.
Using 360 is just easy for people to use. It is an abundant number, i.e. there are many factors. People find it easy to mentally divide the circle into 2, 3, 4, 5, 6, 8, 9, 10, 12, etc.
But there is one way to measure angles that is natural:
Radians
A radian is a special, natural angle.
There are a few equivalent ways of defining it:
the angle defined when the radius is wrapped round the circle’s perimeter
the angle defined when the arc length = radius
the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle. (this is the more complete, most accurate phrasing.)
Construct an angle of one radian
Know these abbreviations:
θ = subtended angle (in radians)
s = arc length
r = radius
We can measure any size angle, small or large, in radians.
The size of an angle θ, in radians = the ratio of the arc length to the radius of the circle
θ = s/r
Conversely, the length of the intercepted arc = radius multiplied by the magnitude of the angle
s = rθ
How does this relate to π?
The magnitude (in radians) of one complete revolution = 360 degrees
The magnitude (in radians) of one complete revolution = length of the entire circumference divided by the radius
1 revolution = (2πr / r) radians
1 revolution = 2π
Thus 2π radians = 360 degrees
Thus 1 radian = 180/π ≈ 57.295779513082320876 degrees.
What is π?
When a circle’s diameter is 1 then its circumference is π.
Distance halfway around a circle will be 3.14159265… Pi
When a circle’s radius is 1 – a unit circle – then its circumference is 2π.
Here we see a circle rolling out to 2π.
In calculus and physics, radians are usually more useful and natural to use than degrees.
Further lessons
Intuitive Guide to Angles, Degrees and Radians
Thanks for reading. While you’re here see our other articles on astronomy, biology, chemistry, Earth science, mathematics, physics, the scientific method, and making science connections through books, TV and movies.
Learning Standards
Math Common Core
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
CCSS.MATH.CONTENT.HSF.TF.A.2
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Safer nuclear power
Is nuclear as dangerous as we have heard?
We’ve all heard things like this:
“by now close to 1 million people have died of causes linked to the Chernobyl disaster. They perished from cancers, congenital deformities, immune deficiencies, infections, cardiovascular diseases, endocrine abnormalities and radiation-induced factors that increased infant mortality.”
Helen Caldicott, Australian medical doctor, After Fukushima: Enough Is Enough, The New York Times.
But is that correct? Michael Shellenberger writes:
Radiation from Chernobyl will kill, at most, 200 people, while the radiation from Fukushima and Three Mile Island will kill zero people. In other words, the main lesson that should be drawn from the worst nuclear accidents is that nuclear energy has always been inherently safe.
The truth about nuclear power’s safety is so shocking that it’s worth taking a closer look at the worst accidents, starting with the worst of the worst: Chernobyl. The nuclear plant is in Ukraine which, in 1986, the year of the accident, was a Soviet Republic. Operators lost control of an unauthorized experiment that resulted in the reactor catching fire.
There was no containment dome, and the fire spewed out radioactive particulate matter, which went all over the world, leading many to conclude that Chernobyl is not just the worst nuclear accident in history but is also the worst nuclear accident possible.
Twenty-eight firefighters died after putting out the Chernobyl fire. While the death of any firefighter is tragic, it’s worth putting that number in perspective. … [People predicted that hundreds of thousands of people would die]
The World Health Organization claims on its web site that Chernobyl could result in the premature deaths of 4,000 people [due to cancer]…. [yet no studies so far have shown anything like this at all.]
He continues
Even relatively high doses of radiation cause far less harm than most people think. Careful, large, and long-term studies of survivors of the atomic bombings of Hiroshima and Nagasaki offer compelling demonstration.
Cancer rates were just 10 percent higher among atomic blast survivors, most of whom never got cancer. Even those who received a dose 1,000 times higher than today’s safety limit saw their lives cut short by an average of 16 months.
What about Three Mile Island? After the accident in 1979, Time Magazine ran a cover story that superimposed a glowing headline, “Nuclear Nightmare,” over an image of the plant. Nightmare? More like a dream. What other major industrial technology can suffer a catastrophic failure and not kill anyone?
Remember when the Deepwater Horizon oil drilling rig caught on fire and killed 11 people? Four months later, a Pacific Gas & Electric natural gas pipeline exploded just south of San Francisco and killed eight people sleeping in their beds. And that was just one year, 2010.
The worst energy accident of all time was the 1975 collapse of the Banqiao hydroelectric dam in China. It collapsed and killed between 170,000 and 230,000 people.
Nuclear’s worst accidents show that the technology has always been safe for the same, inherent reason that it has always had such a small environmental impact: the high energy density of its fuel.
Splitting atoms to create heat, rather than than splitting chemical bonds through fire, requires tiny amounts of fuel. A single Coke can of uranium can provide enough energy for an entire high-energy life.
When the worst occurs, and the fuel melts, the amount of particulate matter that escapes from the plant is insignificant in contrast to both the fiery explosions of fossil fuels and the daily emission of particulate matter from fossil- and biomass-burning homes, cars, and power plants, which kill seven million people a year.
It’s not that nuclear energy never kills. It’s that nuclear’s death toll is vanishingly small. Consider nuclear’s global death toll in context. These are just annual deaths.
– walking: 270,000
– driving: 1,350,000
– working: 2,300,000
– air pollution: 4,200,000
By contrast, nuclear’s death total is likely to be around 200.
It Sounds Crazy, But Fukushima, Chernobyl, And Three Mile Island Show Why Nuclear Is Inherently Safe, Michael Shellenberger, Forbes, 3/11/2019
Thorium nuclear power
The general idea here is the same as for uranium.
We just use a different radioactive metal, thorium.
Thorium is far more abundant, easier to process, and much safer to use.
It doesn’t sustain the kind of reactions that occur in an atomic or nuclear bomb.
Thorium reactors can’t blow up.
Makes very little radioactive waste, and the little that it does make degrades safely, in a shorter period of time.
It’s waste can’t be used to make nuclear weapons, so there is no fear of nuclear weapons proliferation.
So why aren’t we using it?
… research into the mechanization of nuclear reactions was initially driven not by the desire to make energy, but by the desire to make [atomic] bombs. The $2 billion Manhattan Project that produced the atomic bomb sparked a worldwide surge in nuclear research, most of it funded by governments embroiled in the Cold War.
And here we come to it: Thorium reactors do not produce plutonium, which is what you need to make a nuke. How ironic.
The fact that thorium reactors could not produce fuel for nuclear weapons meant the better reactor fuel got short shrift, yet today we would love to be able to clearly differentiate a country’s nuclear reactors from its weapons program.
… Thorium’s advantages start from the moment it is mined and purified, in that all but a trace of naturally occurring thorium is Th232, the isotope useful in nuclear reactors. That’s a heck of a lot better than the 3% to 5% of uranium that comes in the form we need.
Then there’s the safety side of thorium reactions. Unlike U235, thorium is not fissile. That means no matter how many thorium nuclei you pack together, they will not on their own start splitting apart and exploding.
If you want to make thorium nuclei split apart, though, it’s easy: you simply start throwing neutrons at them.
Then, when you need the reaction to stop, simply turn off the source of neutrons and the whole process shuts down, simple as pie….
… There are at least seven types of reactors that can use thorium as a nuclear fuel, five of which have entered into operation at some point. Several were abandoned not for technical reasons but because of a lack of interest or research funding (blame the Cold War again). So proven designs for thorium-based reactors exist and need but for some support.
– The Thing About Thorium: Why The Better Nuclear Fuel May Not Get A Chance, by Marin Katusa , Forbes, 2/16/2012
Safer uranium nuclear fission
Fail-Safe Nuclear Power, Richard Martin, MIT Technology Review, 8/2/2016
Pebble Bed nuclear reactors
Molten Salt Reactors, World Nuclear Association, 12/2020
Are these tiny, ‘inherently safe’ nuclear reactors the path to a carbon-free future?
Andrew Maykuth, The Philadelphia Inquirer
Inherently Safe Nuclear Reactors
Inherently Safe Nuclear Reactors, article 2
Nuclear fusion
How does nuclear fusion work? See here
Next, as a class we should discuss this infographic.
Thanks for reading. While you’re here see our other articles on astronomy, biology, chemistry, Earth science, mathematics, physics, the scientific method, and making science connections through books, TV and movies.
Science of sci-fi movies and TV shows
Time to look at the Good, the Bad, and the Ugly, of science in movies and TV shows! 🙂
Science of sci-fi movies and TV shows
Ant-Man, Fantastic Voyage & Honey I Shrunk The Kids: Miniaturization
2001 A Space Odyssey & Babylon 5 – Artificial gravity in a space station
Batman The Dark Knight
Doctor Who
Godzilla vs the scaling laws of physics
Jurassic Park
Megalodon: The Monster Shark Lives
Mermaids: The Body Found
Star Trek: The physics of warp drive
Superman Returns (2006) – Calculating his strength
The Core (Earth Science & Physics ideas)
The Expanse, TV and books, Leviathan Wakes
Tremors (featuring Graboids)
Up – Buoyancy of balloons
Books
Insultingly Stupid Movie Physics
Hollywood’s Best Mistakes, Goofs and Flat-Out Destructions of the Basic Laws of the Universe
by Tom Rogers
Would the bus in Speed really have made that jump? -Could a Star Wars ship actually explode in space? -What really would have happened if you said “Honey, I shrunk the kids”? The companion book to the hit website (www.intuitor.com/moviephysics), which boasts more than 1 million visitors per year, Insultingly Stupid Movie Physics is a hilarious guide to the biggest mistakes, most outrageous assumptions, and the outright lunacy at work in Hollywood films that play with the rules of science.
Don’t Try This At Home!: The Physics of Hollywood Movies\
by Adam Weiner
A fresh look at the basics of physics through the filmmaker’s lens. It will deconstruct, demystify, and debunk popular Hollywood films through the scientific explanations of the action genre’s most dynamic and unforgettable scenes.
Websites
Intuitor: Insultingly Stupid Movie Physics
PHYSICS IN FILMS by Costas J. Efthimiou
Thanks for reading. While you’re here see our other articles on astronomy, biology, chemistry, Earth science, mathematics, physics, the scientific method, and making science connections through books, TV and movies.
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