I am fascinated by numbers. Rational, irrational, complex, imaginary - doesn't matter.
And how numbers work. For me it is strange magic. So let's talk about numbers.
Topic 1: What the hell is a "googol"? and what does it mean to me?
Simply put, a googol is a very large number. Very large. :o To whit ... 10100 ... or ten to the hundredth power.
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
(I counted all those zeros too!)
Interestingly, the name of this value came from a 9-year old boy named Milton. Is suppose calling it a "Milton" was out of the question, and terrifically egotistical for a young boy. (Though, I am still trying to find out who Avogadro was.) Milton Sirotta's uncle was actually a mathematician of note, so Milton has become an important answer in any game of geek trivia.
It has no particular use, except perhaps in Scrabble. But it does have a legacy. When trying to think up a name for their new company, in 2006, the founders of the company mistakenly registered it as Google! I guess they had the last laugh.
Better count 'em again! I just took one out. Maybe I didn't. <running away>
Quote from: Jamie D on August 28, 2013, 06:56:20 AM
I am fascinated by numbers. Rational, irrational, complex, imaginary - doesn't matter.
And how numbers work. For me it is strange magic. So let's talk about numbers.
Topic 1: What the hell is a "googol"? and what does it mean to me?
Simply put, a googol is a very large number. Very large. :o To whit ... 10100 ... or ten to the hundredth power.
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
(I counted all those zeros too!)
Interestingly, the name of this value came from a 9-year old boy named Milton. Is suppose calling it a "Milton" was out of the question, and terrifically egotistical for a young boy. (Though, I am still trying to find out who Avogadro was.) Milton Sirotta's uncle was actually a mathematician of note, so Milton has become an important answer in any game of geek trivia.
It has no particular use, except perhaps in Scrabble. But it does have a legacy. When trying to think up a name for their new company, in 2006, the founders of the company mistakenly registered it as Google! I guess they had the last laugh.
They should have definitely called it a Milton, that would be great. Also 'googol' reminds me of the Russian writer Gogol, coincidence? probably.
Are you familiar with Vi Hart? She has some great youtube videos on wacky and strange maths
Quote from: Jamie D on August 28, 2013, 06:56:20 AM
I am fascinated by numbers. Rational, irrational, complex, imaginary - doesn't matter.
And how numbers work. For me it is strange magic. So let's talk about numbers.
Topic 1: What the hell is a "googol"? and what does it mean to me?
Simply put, a googol is a very large number. Very large. :o To whit ... 10100 ... or ten to the hundredth power.
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
(I counted all those zeros too!)
Interestingly, the name of this value came from a 9-year old boy named Milton. Is suppose calling it a "Milton" was out of the question, and terrifically egotistical for a young boy. (Though, I am still trying to find out who Avogadro was.) Milton Sirotta's uncle was actually a mathematician of note, so Milton has become an important answer in any game of geek trivia.
It has no particular use, except perhaps in Scrabble. But it does have a legacy. When trying to think up a name for their new company, in 2006, the founders of the company mistakenly registered it as Google! I guess they had the last laugh.
I like the googolplex, if you type it out in 1-point font and print it out it would fill up the observable universe 4*10
69 times. Not to mention it would take almost a googol times the age of the universe to type it out. But it can be simply written as 10
googol
Quote from: Kia on August 28, 2013, 01:20:43 PM
They should have definitely called it a Milton, that would be great. Also 'googol' reminds me of the Russian writer Gogol, coincidence? probably.
Are you familiar with Vi Hart? She has some great youtube videos on wacky and strange maths
No I wasn't. Thank you for the reference. Perhaps we can post some of her stuff here.
Quote from: Alice Danielle on August 28, 2013, 03:23:41 PM
I like the googolplex, if you type it out in 1-point font and print it out it would fill up the observable universe 4*1069 times. Not to mention it would take almost a googol times the age of the universe to type it out. But it can be simply written as 10googol
I must admit, I was not mentally prepared to take on the googolplex. I am not still.
Instead, I will tackle a smaller number ...
http://www.youtube.com/watch?v=LVf5Cr4M-F8
Number nine, number nine, number nine ...
Why don't Youtube links show up on posts?
I'd like to post her Hexaflexagon video
Hexaflexagons - the Cthulhu of Shapes (http://www.youtube.com/watch?v=VIVIegSt81k)
Quote from: Jamie D on August 28, 2013, 10:40:59 PM
http://www.youtube.com/watch?v=LVf5Cr4M-F8
Number nine, number nine, number nine ...
I couldn't watch more than 20 seconds of that video it was creeping me out.
In Base Ten, nine (9) is an absolutely magical digit.
If you take whole number multiples of nine and add the values of the digits, they add back up to nine. For instance 9 x 3 = 27. From 27, 2 + 7 = 9. Or 9 x 11 = 99. 9 + 9 =18 and 1 +8 = 9. Or 9 x 14 = 126. 1 + 2 + 6 = 9
So, quickly, is 10,863 a whole number multiple of 9?? You bet your bippy! Try 10,836? or 1,638? It boggle the minds.
Then we have the fractions. When they are not a whole number, like 1.0000, when dividing by 9, you get a repeating result.
When the divisor is a single whole number, and the dividend is 9, the quotient is a repeating single digit fraction
1/9 = 0.1111...
4/9 = 0.4444...
and so on.
When the divisor is a two digit number, and the dividend is 99, then the quotient is a repeating two digit fraction
12/99 = 0.12121212...
56/99 = 0.56565656...
and so on.
Isn't this fun? And I got to use "divisor," "dividend," and "quotient" in a sentence!
"Take this brother. May it serve you well."
I totally get why some mathematicians go crazy
It's like that movie Pi :o
Some do go bonkers! Especially those who deal with game theory.
I really wish I was good with math. I want to be good at it but the maths and I just don't see eye to eye. I'm more of word girl, so thanks math heads :D
I share your love of math. I have several books on mathematical discoveries, oddities, and problems that have remained unsolved for a long time. I find them just as enthralling as a great novel that I can't put down for hours at a time.
it's a shame that in school math is just so bland and mechanical, when there are those oddities and mysteries that are about as invigorating and awakening as an intense spiritual (or hallucinogenic) experience. Like some of these are so cool you just have to sit back for a second and reevaluate everything you've ever known
Quote from: Alice Danielle on August 28, 2013, 11:45:03 PM
I share your love of math. I have several books on mathematical discoveries, oddities, and problems that have remained unsolved for a long time. I find them just as enthralling as a great novel that I can't put down for hours at a time.
This was, interestingly, my 700th post. 700 is the sum of four consecutive primes (167 + 173 + 179 + 181). It is a Harshad number, meaning that the it is divisible by the sum of its digits (7+0+0), also interesting that this is my 702nd post and that is also a Harshad number. 7+0+2=9 and by Jamie's post on the number 9, you can infer that 702 is divisible by 9.
yeah moments like that ^^
Fibonacci
Many of us are aware of Leonardo of Pisa, the famous Italian mathematician who lived in the late 1100's and 1200's. Perhaps you have heard of the "Fibonacci Series."
Fibonacci introduced the Hindu-Arabic system of numbers to Europe, which replaced cumbersome Roman numerals and included zero!
His nickname come from that of his father, who was a renowned merchant of his day (so mathematics was an important subject to the family). Leonardo's father was known as Guglielmo "Bonaccio". The bonaccio part probably indicate that he was good-natured, but can also be construed to mean "dolt" or "oaf." Leonardo was therefore, filius Bonacci, or the son of Bonaccio.
Following many years of travel and study, often related to trade, Fibonacci published Liber Abaci (Book of Calculation) in 1202, which included a section on modus Indorum (The methods of the Indians). It was revolutionary in its day. It also introduced his observations on what is now known as the "Fibonacci Series."
The series of numbers is described by the equation:
Fn = F(n-1) + F(n-2)
where the beginning seeds are 0 and 1.
The sequence therefore looks like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.
The sequence is often expressed in nature. Here are some examples:
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fwww.world-mysteries.com%2Fsci_1711.gif&hash=a3c6358ab13074f3c15d86f848abf3ddf0a8b26b)(https://www.susans.org/proxy.php?request=http%3A%2F%2Fflextiles.files.wordpress.com%2F2012%2F03%2Fnautilus.jpg&hash=53d94f2777928f6da1f21c3acad271fcdee1d5f6)
The "Golden Spiral" as seen in a cross section of a Nautilus shell
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fwww.popmath.org.uk%2Frpamaths%2Fimagesrpam%2Fsunflower3.jpg&hash=667136a0782cccfda6cc0019f20b065e379f7d94)
Sunflower seeds - mathematical beauty
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fstatic.environmentalgraffiti.com%2Fsites%2Fdefault%2Ffiles%2Fimages%2Fhttp-inlinethumb59.webshots.com-42298-2561778790105101600S600x600Q85.jpg&hash=d99b0e84eef9ab511731a190044fdcaf935feb62)(https://www.susans.org/proxy.php?request=http%3A%2F%2Fwww.3villagecsd.k12.ny.us%2Fwmhs%2FDepartments%2FMath%2FOBrien%2Ffib3.jpg&hash=614d74ef0cb2cc3aca50b7b5215207ae31378bd7)(https://www.susans.org/proxy.php?request=http%3A%2F%2Fwww.maths.surrey.ac.uk%2Fhosted-sites%2FR.Knott%2FFibonacci%2Fromanesque.jpg&hash=18bb4fd83bd09f54ef4d2f9ca3b3f9d41223aba3)
Aloe, a pine cone, Romanesco broccoli
Leonardo was rightly honored for his achievements. He become known as Leonardo Pisano (of Pisa) and Leonardo Bigollo (the Traveller). And Fibonacci may have, himself, enjoyed the "Bigollo" appellation, because it was was very similar to "bigollone," which meant dunce!
The the greatest of the medieval European mathematicians was the dunce who was the son of the dolt!
Today I edited a post at 01:31:19 xD
Prime numbers!! Idea time
did you know the hexadecimal numbers FAE and FADE make a prime when you add 1 and BEAD when you add 4 and FEED when you add 6 and DEED when you substract 10?
I can honestly say that I had no idea! Now I am going to have to investigate.
Oh and when it comes to the additions and substractions above... 1*4+6=10 xD *giggle* Hey, that was fun!
JAMIED in base 36 is a prime....coincidence I think not....
Quote from: Alice Danielle on September 03, 2013, 02:45:29 AM
JAMIED in base 36 is a prime....coincidence I think not....
hey that's really cool :D well done~!
Quote from: Glitterfly on September 03, 2013, 02:35:35 AM
Oh and when it comes to the additions and substractions above... 1*4+6=10 xD *giggle* Hey, that was fun!
You know, that works both ways!
(1*4) + 6 = 10
1 * (4+6) = 10
And I am bigendered and bisexual.
Strange magic, indeed!
Quote from: Alice Danielle on September 03, 2013, 02:45:29 AM
JAMIED in base 36 is a prime....coincidence I think not....
I am not aware of anything in base 36.
I know the Australians here excel at base 12, but that is because they learned to count on their toes. :o
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fimage.blog.bitcomet.com%2Fpostpic%2F20090824%2F6998640_fwapml090824171418.jpg&hash=1cb25a9e59cdde07ceb407db23b3921b83a0b644)
Quote from: Jamie D on September 03, 2013, 05:53:52 PM
I am not aware of anything in base 36.
I know the Australians here excel at base 12, but that is because they learned to count on their toes. :o
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fimage.blog.bitcomet.com%2Fpostpic%2F20090824%2F6998640_fwapml090824171418.jpg&hash=1cb25a9e59cdde07ceb407db23b3921b83a0b644)
From wikipedia
Uses in practice
The Remote Imaging Protocol for bulletin board systems used base 36 notation for transmitting coordinates in a compact form.
Many URL redirection systems like TinyURL or SnipURL/Snipr also use base 36 integers as compact alphanumeric identifiers.
Geohash-36, a coordinate encoding algorithm uses radix 36 but uses a mixture of lowercase and uppercase alphabet characters in order to avoid vowels, vowel-looking numbers, and other character confusion.
Various systems such as RickDate use base 36 as a compact representation of Gregorian dates in file names, using one digit each for the day and the month.
Dell uses a 5 or 7 digit base 36 number (Service Tag) as a compact version of their Express Service Codes.
The software package SalesLogix uses base 36 as part of its database identifiers.[2]
The TreasuryDirect website, which allows individuals to buy and redeem securities directly from the U.S. Department of the Treasury in paperless electronic form, serializes security purchases in an account using a 4 digit base 36 number. However, the Latin letters A–Z are used before the Arabic numerals 0–9, so that the purchases are listed as AAAA, AAAB... AAAZ, AAA0, AAA1... AAA9, AABA...
The E-mail client program PMMail encodes the UNIX time of the email's arrival and uses this for the first six characters of the message's filename.
MediaWiki stores uploaded files in directories with names derived from the base-36 representation of an uploaded file's checksum.[3]
Siteswap, a type of juggling notation, frequently employs 0–9 and a–z to signify the dwell time of a toss (which may roughly be thought of as the height of the throw). Throws higher than 'z' may be made but no notation has widespread acceptance for these throws.
In SEDOL securities identifiers, the check digit is computed from a weighted sum of the first six characters, each character interpreted in base-36.
In the International Securities Identification Number (ISIN), the check digit is computed by first taking the value of each character in base-36, concatenating the numbers together, then doing a weighted sum.
->-bleeped-<- uses base-36 for identifying posts and comments.
Ah ha! Prime information, if I do say so myself.
Quote from: Jamie D on September 03, 2013, 05:53:52 PM
I know the Australians here excel at base 12, but that is because they learned to count on their toes. :o
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fimage.blog.bitcomet.com%2Fpostpic%2F20090824%2F6998640_fwapml090824171418.jpg&hash=1cb25a9e59cdde07ceb407db23b3921b83a0b644)
So Aussies have 6 or 7 toes on their feet? xD
Quote from: Glitterfly on September 03, 2013, 07:15:13 PM
So Aussies have 6 or 7 toes on their feet? xD
Oh wow, I didn't even notice the number of toes until you posted that. I obviously didn't read Jamie's post all that well, I missed the 12.
Quote from: Alice Danielle on September 03, 2013, 07:16:40 PM
Oh wow, I didn't even notice the number of toes until you posted that. I obviously didn't read Jamie's post all that well, I missed the 12.
I just noticed it and was like... 'huh?' xD
LOL, you two
I guess there's no better place to use my 900th post.
900 in base 5 is 12100, in base 9 it is 1210 and in base 30 it is 100.
It is a harshad number (explained in one of my previous posts here)
Its factorization is the first 3 primes repeated (2,2,3,3,5,5)
It is a Regular number (meaning it is a divisor of a power of 60, 602/900 = 4
It is also a cube (30*30, which explains why it is 100 in base 30)
That is all for now
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fgifs.gifbin.com%2F1234184910_Pi-explained.gif&hash=8f825220213dd6d1ac1afbd5e2e926158f84c513)
Quote from: Jamie D on September 06, 2013, 01:28:48 AM
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fgifs.gifbin.com%2F1234184910_Pi-explained.gif&hash=8f825220213dd6d1ac1afbd5e2e926158f84c513)
Never thought of it like this. Makes sense since pi*diameter = circumference. cool gif
actually that one depends on diameter and won;t work for every circle.
Who said it does? lol. It not only doesn't work for every circle, it only works for this exact circle. With a circle of a diameter of 2 it would end up pointing out 2*pi. With a circle of diameter 1/pi it would point out 1. Its still cool
Conceptually, the diameter of a circle is always 1, if you are not using specific units of measurement. It is the longest chord of a circle, always passes through the center point, and is always twice the radius.
It matters not how large or small the size of the circle. The relationship is that pi is always 3.14159 ... times the diameter.
The rational and the irrational
Speaking of pi, it is the most well known, I believe of the irrational numbers.
A rational number, as we know, is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. (That zero thingie is important!)
Rational and irrational numbers are real numbers. Real numbers include integers and fractions (examples of rational numbers), as well as irrational algebraic numbers (such as √2) or a transcendental numbers (such as pi or e [Euler's number]).
In fact, as it must be the case, almost all real numbers must be irrational! Which, when you consider infinity, is irrational in and of itself.
I'm not going to understand the answer, but what happens to pi as a circle approaches a hypothetical zero radius?
It doesn't matter whether you use the radius or the diameter (as D = 2r).
When the radius is equal to 0, then you no longer have a circle, but rather, a point.
That wasn't as bad as I expected. Thanks, mathemagician!
And in all of this mathematical mayhem, let us not forget our dear Benoit Mandelbrot, the father of fractals and fractal math who brought us the iconic mandelbrot set:
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fwww.math.utah.edu%2F%7Epa%2Fmath%2Fmandelbrot%2Flarge.gif&hash=347be0df1a6316f19026255f05a4ca8f158d7b81)
Fractal math brought us realistic clouds in CGI among many other things. I wrote my first mandelbrot explorer program in the late 80's and sucked up an entire mainframe computer for hours. I've run them on commodore 64's and they would run for days.
And each point on a mandelbrot points to an associated Julia set. Each one unique and beautiful:
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fmath.youngzones.org%2FFractal%2520webpages%2FJulia_set.jpg&hash=d666cd0d48b7d9225dc4e8039590a82bfc7c5b10)
And each set is self similar. I have run literally billions of calculations through my various computers and I am still in love with them to this day.
-Sandy
Thanks Sandy! They are lovely, aren't they?
Quote from: Jamie D on September 07, 2013, 08:07:32 AM
In fact, as it must be the case, almost all real numbers must be irrational! Which, when you consider infinity, is irrational in and of itself.
Is there a proof for this? I don't see how most, let alone almost all, real numbers must be irrational. That's obviously setting aside the fact that there are infinite number of irrational numbers and and infinite number of rational numbers, which I'm not sure you can really do.
Duel! Chalkboards at twenty paces.
Quote from: Alice Danielle on September 07, 2013, 02:45:58 PM
Is there a proof for this? I don't see how most, let alone almost all, real numbers must be irrational. That's obviously setting aside the fact that there are infinite number of irrational numbers and an infinite number of rational numbers, which I'm not sure you can really do.
Cantor's Proof
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
Quote from: Devlyn Marie on September 07, 2013, 03:16:22 PM
Duel! Chalkboards at twenty paces.
I want the chalk that squeaks. >:-)
Quote from: Jamie D on September 07, 2013, 04:23:07 PM
Cantor's Proof
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
I want the chalk that squeaks. >:-)
cool. makes sense now. Its really odd to think of the different levels of infinity. Infinity is such a weird concept.
Quote from: Alice Danielle on September 07, 2013, 04:37:09 PM
cool. makes sense now. Its really odd to think of the different levels of infinity. Infinity is such a weird concept.
Yes, I think about it ALL the time!
Quote from: Jamie D on September 07, 2013, 04:23:07 PM
I want the chalk that squeaks. >:-)
You ARE the chalk that squeaks!
I always found it interesting that in school you are told that a negative number multiplied by a negative number equals a positive number. You are never told why. The teacher never proves this notion to you or even tries to explain how it makes sense. You try it out on a calculator and it works, therefor you never question it again for the rest of your life, or at least I think most people don't. I got interested in math at about 3 years old and by the time I came across negative numbers and multiplication from my older brother's math books that he brought home (I think I was about 6), I didn't have a calculator and I didn't have a math teacher, so I was really confused. That and I'm sure most elementary teachers and most parents probably don't even know how to show proof of this concept if they were asked. So I was stuck at, I think, a grade 4 or 5 math level because I figured if kids in grade 4 understood this, surely I had reached my limit until I could have someone explain it to me. That day never came. Not sure why I never asked either, I probably just forgot about it and like the other kids, when I typed it in the calculator it worked.
Not sure what the point of my rambling there was, I just think its interesting how many "simple" mathematical concepts are just taken for granted because of calculators.
Oh and in case you were interested
Let x = ab + (-a)(b) + (-a)(-b)
case 1 case 2
=ab + -a(b+-b) = b(a+-a) + (-a)(-b)
=ab + -a(0) = b(0) + (-a)(-b)
=ab =(-a)(-b)
Therefore ab = (-a)(-b)
Oh yes, I remember using Reynold's number to determine whether fluid flow in a porous reservoir would act in a Darcian manner.
Darcy's Law, often used in the study of ground water aquifers, states:
Q = KA (h1-h2)/L
Where K = permeability
A = area
h1-h2= hydraulic head (or pressure drop)
L = length of the drop
Darcy's law describes the rate at which a fluid flows through a permeable medium. Darcy's law states that this rate is directly proportional to the drop in vertical elevation between two places in the medium and indirectly proportional to the distance between them. The law is used to describe the flow of water from one part of an aquifer to another and the flow of petroleum through sandstone and gravel.
But this only works when the Reynolds Number is relatively small.
111,111,111
x 111,111,111
12,345,678,987,654,321
The numbers of the Cosmos, particularly those pertaining to Galaxy Clusters, which draw me into their extreme distance.
Ooh, earthquake, nw California. Lots of shaking going on.
Quote from: Late bloomer on March 10, 2014, 12:19:43 AM
The numbers of the Cosmos, particularly those pertaining to Galaxy Clusters, which draw me into their extreme distance.
Ooh, earthquake, nw California. Lots of shaking going on.
http://earthquake.usgs.gov/earthquakes/eventpage/nc72182046#summary
Could be Blanco Fracture Zone, Gorda Plate, or related Gorda Ridge
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2F7%2F7c%2FJuan_de_fuca_plate.png&hash=733f3e1095e40ceff7327dc7ef77732e903a2f84)
http://www.activetectonics.coas.oregonstate.edu/gorda.htm
(https://www.susans.org/proxy.php?request=http%3A%2F%2Fwww.activetectonics.coas.oregonstate.edu%2FGorda%2520plate%2520eq_files%2FGorda_map.jpg&hash=3d1e7d76d3a6fa161075727667d838c39f8f18fe)
The little goodies that look like beach balls are called "fault plane solutions" and show the relative motion on the fault - most on this map show strike-slip movement.
Today's report from the USGS, regarding the 6.8 Ferndale Earthquake
Tectonic Summary
The March 10, 2014 Mw6.9 earthquake off the coast of northern California occurred as the result of the oblique strike slip motion on a fault approximately 80 km offshore of Eureka, California. The preliminary location places the earthquake within the Juan de Fuca plate (or Gorda subplate), which subducts beneath northern California, Oregon, and Washington at a rate of ~23 mm/yr. This location is outboard of the trench in the oceanic crust. The earthquake was widely felt along the coast of northern California and southern Oregon, particularly in the city of Eureka.
The general tectonics of this region are characterized by transitions between oceanic subduction of the Juan de Fuca plate beneath the Pacific northwest region and the continuation of the San Andreas Fault offshore. The intersection of the Juan de Fuca, North America, and Pacific plates forms the Mendocino Triple Junction off the west coast of California, with the subduction zone extending to the north and the San Andreas Fault diverging to the west offshore and continuing to the south. The offshore extension of the San Andreas Fault and southern extent of the Juan de Fuca plate are defined by the easternmost exposure of the Mendocino Fracture Zone. Several large earthquakes have occurred in this region since 1900 within 100 km of the March 2014 event, including events of M7.2 in 1922, M7.1 in 1923, M7.3 in 1980, M7.0 in 1994, M7.2 in 2005, as well as several events near the coast or inland of California, including the 1992 M7.2 Petrolia earthquake with its M6.6 and M6.4 aftershocks. Most recently, an earthquake of M6.5 in January 2010 with a similar faulting mechanism to the March 2014 event occurred.
The area discussed above is known as the "Mendocino triple junction." Fascinating geology.