$21 Sponge Holder/Sink Caddy Ceramic Pottery, Stone with Purple Blue Handmade Products Home Kitchen Home Décor Caddy,Stone,Pottery,,Blue,Holder/Sink,Purple,with,Handmade Products , Home Kitchen , Home Décor,$21,www.happykidsart.nl,Sponge,/facular586331.html,Ceramic Sponge Holder Sink Caddy Ceramic Blue Purple Pottery with Outlet sale feature Stone Sponge Holder Sink Caddy Ceramic Blue Purple Pottery with Outlet sale feature Stone $21 Sponge Holder/Sink Caddy Ceramic Pottery, Stone with Purple Blue Handmade Products Home Kitchen Home Décor Caddy,Stone,Pottery,,Blue,Holder/Sink,Purple,with,Handmade Products , Home Kitchen , Home Décor,$21,www.happykidsart.nl,Sponge,/facular586331.html,Ceramic

Sponge Time sale Holder Sink Caddy Ceramic Blue Purple Pottery with Outlet sale feature Stone

Sponge Holder/Sink Caddy Ceramic Pottery, Stone with Purple Blue


Sponge Holder/Sink Caddy Ceramic Pottery, Stone with Purple Blue


This handy little pot keeps your sponge high and dry, and looks pretty. Beautiful and useful. Also stores scouring pads easily.

Sponge included.

To view a variety of color choices, visit me shop on Amazon Handmade.

Sponge Holder/Sink Caddy Ceramic Pottery, Stone with Purple Blue

Earth System Models simulate the changing climate

Image credit: NASA.

The climate is changing, and we need to know what changes to expect and how soon to expect them. Earth system models, which simulate all relevant components of the Earth system, are the primary means of anticipating future changes of our climate [TM219 or search for “thatsmaths” at SPEARFISHING WORLD Pole Spear Shrink Grip - Glue Lined].

WINTOOLS 8 PCS Petrol Engine Cylinder Compression Tester Kit Aut

The Signum Function may be Continuous

Abstract: Continuity is defined relative to a topology. For two distinct topological spaces and having the same underlying set but different families of open sets, a function may be continuous in one but discontinuous in the other. Continue reading ‘The Signum Function may be Continuous’

The Social Side of Mathematics

On a cold December night in 1976, a group of mathematicians assembled in a room in Trinity College Dublin for the inaugural meeting of the Irish Mathematical Society (IMS). Most European countries already had such societies, several going back hundreds of years, and it was felt that the establishment of an Irish society to promote the subject, foster research and support teaching of mathematics was timely [TM218 or search for “thatsmaths” at Tackle Rigs Fishing Leaders Stainless Steel Wire Trace Leader 2].

Continue reading ‘The Social Side of Mathematics’

Real Derivatives from Imaginary Increments

The solution of many problems requires us to compute derivatives. Complex step differentiation is a method of computing the first derivative of a real function, which circumvents the problem of roundoff error found with typical finite difference approximations.

Rounding error and formula error as functions of step size [Image from Wikimedia Commons].

For finite difference approximations, the choice of step size is crucial: if is too large, the estimate of the derivative is poor, due to truncation error; if is too small, subtraction will cause large rounding errors. The finite difference formulae are ill-conditioned and, if is very small, they produce zero values.

Where it can be applied, complex step differentiation provides a stable and accurate method for computing .

Continue reading ‘Real Derivatives from Imaginary Increments’

Changing Views on the Age of the Earth

[Image credit: NASA]

In 1650, the Earth was 4654 years old. In 1864 it was 100 million years old. In 1897, the upper limit was revised to 40 million years. Currently, we believe the age to be about 4.5 billion years. What will be the best guess in the year 2050? [TM217 or search for “thatsmaths” at Mooto Korea Taekwondo Reversible Chest Guard (1 EA) Approved Pro].

Continue reading ‘Changing Views on the Age of the Earth’

Carnival of Mathematics

The Aperiodical is described on its `About’ page as “a meeting-place for people who already know they like maths and would like to know more”. The Aperiodical coordinates the Carnival of Mathematics (CoM), a monthly blogging roundup hosted on a different blog each month. Generally, the posts describe a collection of interesting recent items on mathematics from around the internet. This month, it is the turn of thatsmaths.com to host CoM.
Continue reading ‘Carnival of Mathematics’

Phantom traffic-jams are all too real

Driving along the motorway on a busy day, you see brake-lights ahead and slow down until the flow grinds to a halt. The traffic stutters forward for five minutes or so until, mysteriously, the way ahead is clear again. But, before long, you arrive at the back of another stagnant queue. Hold-ups like this, with no apparent cause, are known as phantom traffic jams and you may experience several such delays on a journey of a few hours [TM216 or search for “thatsmaths” at Victor 6500 12-Digit Desktop Financial Calculator, Loan Mortga].

Traffic jams can have many causes [Image © Susanneiles.com. JPEG]

Continue reading ‘Phantom traffic-jams are all too real’

Simple Models of Atmospheric Vortices

Atmospheric circulation systems have a wide variety of structures and there is no single mechanistic model that describes all their characteristics. However, we can construct simple kinematic models that capture some primary aspects of the flow. For simplicity, we will concentrate on idealized extra-tropical depressions. We will not consider hurricanes and tropical storms in any detail, because the effects of moisture condensation and convection dominate their behaviour.

Continue reading ‘Simple Models of Atmospheric Vortices’

Finding Fixed Points

An isometry on a metric space is a one-to-one distance-preserving transformation on the space. The Euclidean group is the group of isometries of -dimensional Euclidean space. These are all the transformations that preserve the distance between any two points. The group depends on the dimension of the space. For the Euclidean plane , we have the group , comprising all combinations of translations, rotations and reflections of the plane.

Continue reading ‘Finding Fixed Points’

All Numbers Great and Small

Is space continuous or discrete? Is it smooth, without gaps or discontinuities, or granular with a limit on how small a distance can be? What about time? Can time be repeatedly divided into smaller periods without any limit, or is there a shortest interval of time? We don’t know the answers. There is much we do not know about physical reality: is the universe finite or infinite? Are space and time arbitrarily divisible? Does our number system represent physical space and time? [TM215 or search for “thatsmaths” at Face Scrub/Polish - Foaming Natural Grapeseed, Honey-Granules Fa]. Continue reading ‘All Numbers Great and Small’

Approximating the Circumference of an Ellipse

The realization that the circumference of a circle is related in a simple way to the diameter came at an early stage in the development of mathematics. But who was first to prove that all circles are similar, with the ratio of circumference to diameter the same for all? Searching in Euclid’s Elements, you will not find a proof of this. It is no easy matter to define the length of a curve? It required the genius of Archimedes to prove that is constant, and he needed to introduce axioms beyond those of Euclid to achieve this; see earlier post here.

Continue reading ‘Approximating the Circumference of an Ellipse’

Kalman Filters: from the Moon to the Motorway

Before too long, we will be relieved of the burden of long-distance driving. Given the desired destination and access to a mapping system, electronic algorithms will select the best route and control the autonomous vehicle, constantly monitoring and adjusting its direction and speed of travel. The origins of the methods used for autonomous navigation lie in the early 1960s, when the space race triggered by the Russian launch of Sputnik I was raging  [TM214 or search for “thatsmaths” at Aqua Lung Thermocline Flex Glove - Mens 5MM- XL].

Continue reading ‘Kalman Filters: from the Moon to the Motorway’

Gauss Predicts the Orbit of Ceres

Ceres (bottom left), the Moon and Earth, shown to scale [Image NASA].

On the first day of a new century, January 1, 1801, astronomer Giuseppe Piazzi discovered a new celestial object, the minor planet Ceres. He made about 20 observations from his observatory in Palermo before the object was lost in the glare of the Sun in early February. Later in the year, several astronomers tried without success to locate it. Without accurate knowledge of its orbit, the search seemed hopeless. How could its trajectory be determined from a few observations made from the Earth, which itself was moving around the Sun?

Continue reading ‘Gauss Predicts the Orbit of Ceres’

Seeing beyond the Horizon

From a hilltop, the horizon lies below the horizontal level at an angle called the “dip”. Around AD 1020, the brilliant Persian scholar al-Biruni used a measurement of the dip, from a mountain of known height, to get an accurate estimate of the size of the Earth. It is claimed that his estimate was within 1% of the true value but, since he was not aware of atmospheric refraction and made no allowance for it, this high precision must have been fortuitous  [TM213 or search for “thatsmaths” at Mallory SC616NP TRANSDUCER, PIEZO, 2.9KHZ, 95DBA, 16VDC].

Snowdonia photographed from the Ben of Howth, 12 January 2021. Photo: Niall O’Carroll (Instagram).

Continue reading ‘Seeing beyond the Horizon’

Al Biruni and the Size of the Earth

Abu Rayhan al-Biruni (AD 973–1048)

Al Biruni at Persian Scholars Pavilion in Vienna.

The 11th century Persian mathematician Abu Rayhan al-Biruni used simple trigonometric results to estimate the radius and circumference of the Earth. His estimate has been quoted as 6,340 km, which is within 1% of the mean radius of 6,371 km. While al-Biruni’s method was brilliant and, for its era, spectacular, the accuracy claimed must be regarded with suspicion.

Al-Biruni assumed that the Earth is a perfect sphere of (unknown) radius . He realised that because of the Earth’s curvature the horizon, as viewed from a mountain-top, would appear to be below the horizontal direction. This direction is easily obtained as being orthogonal to the vertical, which is indicated by a plumb line.

Continue reading ‘Al Biruni and the Size of the Earth’

The Simple Arithmetic Triangle is full of Surprises

Pascal’s triangle is one of the most famous of all mathematical diagrams, simple to construct and yet rich in mathematical patterns. These can be found by a web search, but their discovery by study of the diagram is vastly more satisfying, and there is always a chance of finding something never seen before  [TM212 or search for “thatsmaths” at Fidragon Mlock Rail 5-Slot 9-Slot 11-Slot 16 pcs mlock Screw and].

Pascal’s triangle as found in Zhu Shiji’s treatise The Precious Mirror of the Four Elements (1303).

Continue reading ‘The Simple Arithmetic Triangle is full of Surprises’

Hanoi Graphs and Sierpinski’s Triangle

The Tower of Hanoi is a famous mathematical puzzle. A set of disks of different sizes are stacked like a cone on one of three rods, and the challenge is to move them onto another rod while respecting strict constraints:

  • Only one disk can be moved at a time.
  • No disk can be placed upon a smaller one.

Tower of Hanoi [image Wikimedia Commons].

Continue reading ‘Hanoi Graphs and Sierpinski’s Triangle’

Multi-faceted aspects of Euclid’s Elements

A truncated octahedron within the coronavirus [image from Cosico et al, 2020].

Euclid’s Elements was the first major work to organise mathematics as an axiomatic system. Starting from a set of clearly-stated and self-evident truths called axioms, a large collection of theorems is constructed by logical reasoning. For some, the Elements is a magnificent triumph of human thought; for others, it is a tedious tome, painfully prolix and patently pointless  [TM211 or search for “thatsmaths” at omyBigDeal Non-Stick Sewing Presser Foot Industrial Needle Machi]. Continue reading ‘Multi-faceted aspects of Euclid’s Elements’

A Model for Elliptic Geometry

For many centuries, mathematicians struggled to derive Euclid’s fifth postulate as a theorem following from the other axioms. All attempts failed and, in the early nineteenth century, three mathematicians, working independently, found that consistent geometries could be constructed without the fifth postulate. Carl Friedrich Gauss (c. 1813) was first, but he published nothing on the topic. Nikolai Ivanovich Lobachevsky, around 1830, and János Bolyai, in 1832, published treatises on what is now called hyperbolic geometry.

Continue reading ‘A Model for Elliptic Geometry’

Improving Weather Forecasts by Reducing Precision

Weather forecasting relies on supercomputers, used to solve the mathematical equations that describe atmospheric flow. The accuracy of the forecasts is constrained by available computing power. Processor speeds have not increased much in recent years and speed-ups are achieved by running many processes in parallel. Energy costs have risen rapidly: there is a multimillion Euro annual power bill to run a supercomputer, which may consume something like 10 megawatts [TM210 or search for “thatsmaths” at R.S. Nok - EZ and Snap - on String Combo].

The characteristic butterfly pattern for solutions of Lorenz’s equations [Image credit: source unknown].

Continue reading ‘Improving Weather Forecasts by Reducing Precision’

Can You Believe Your Eyes?

Scene from John Ford’s Stagecoach (1939).

Remember the old cowboy movies? As the stage-coach comes to a halt, the wheels appear to spin backwards, then forwards, then backwards again, until the coach stops. How can this be explained?

Continue reading ‘Can You Believe Your Eyes?’

The Size of Things

In Euclidean geometry, all lengths, areas and volumes are relative. Once a unit of length is chosen, all other lengths are given in terms of this unit. Classical geometry could determine the lengths of straight lines, the areas of polygons and the volumes of simple solids. However, the lengths of curved lines, areas bounded by curves and volumes with curved surfaces were mostly beyond the scope of Euclid. Only a few volumes — for example, the sphere, cylinder and cone — could be measured using classical methods.

Continue reading ‘The Size of Things’

Entropy and the Relentless Drift from Order to Chaos

In a famous lecture in 1959, scientist and author C P Snow spoke of a gulf of comprehension between science and the humanities, which had become split into “two cultures”. Many people in each group had a lack of appreciation of the concerns of the other group, causing grave misunderstandings and making the world’s problems more difficult to solve. Snow compared ignorance of the Second Law of Thermodynamics to ignorance of Shakespeare [TM209 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Entropy and the Relentless Drift from Order to Chaos’

Circles, polygons and the Kepler-Bouwkamp constant

If circles are drawn in and around an equilateral triangle (a regular trigon), the ratio of the radii is . More generally, for an N-gon the ratio is easily shown to be . Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular orbits separated by regular polygons (see earlier post on the Mysterium Cosmographicum here).

Kepler was unable to construct an accurate model using polygons, but he noted that, if successive polygons with an increasing number of sides were inscribed within circles, the ratio did not diminish indefinitely but appeared to tend towards some limiting value. Likewise, if the polygons are circumscribed, forming successively larger circles (see Figure below), the ratio tends towards the inverse of this limit. It is only relatively recently that the limit, now known as the Kepler-Bouwkamp constant, has been established. 

Continue reading ‘Circles, polygons and the Kepler-Bouwkamp constant’

Was Space Weather the cause of the Titanic Disaster?

Space weather, first studied in the 1950’s, has grown in importance with recent technological advances. It concerns the influence on the Earth’s magnetic field and upper atmosphere of events on the Sun. Such disturbances can enhance the solar wind, which interacts with the magnetosphere, with grave consequences for navigation. Space weather affects the satellites of the Global Positioning System, causing serious navigation problems [TM208 or search for “thatsmaths” at irishtimes.com].

Solar disturbances disrupt the Earth’s magnetic field [Image: ESA].
Continue reading ‘Was Space Weather the cause of the Titanic Disaster?’

The Dimension of a Point that isn’t there

A slice of Swiss cheese has one-dimensional holes;
a block of Swiss cheese has two-dimensional holes.

What is the dimension of a point? From classical geometry we have the definition “A point is that which has no parts” — also sprach Euclid. A point has dimension zero, a line has dimension one, a plane has dimension two, and so on.

Continue reading ‘The Dimension of a Point that isn’t there’

Making the Best of Waiting in Line

Queueing system with several queues, one for each serving point [Wikimedia Commons].

Queueing is a bore and waiting to be served is one of life’s unavoidable irritants. Whether we are hanging onto a phone, waiting for response from a web server or seeking a medical procedure, we have little choice but to join the queue and wait. It may surprise readers that there is a well-developed mathematical theory of queues. It covers several stages of the process, from patterns of arrival, through moving gradually towards the front, being served and departing  [TM207 or search for “thatsmaths” at Timer Relay Glow Plug 31A66-15100 For Tractor 7000 7200 7205 726].

Continue reading ‘Making the Best of Waiting in Line’

Differential Forms and Stokes’ Theorem

Elie Cartan (1869–1951).

The theory of exterior calculus of differential forms was developed by the influential French mathematician Élie Cartan, who did fundamental work in the theory of differential geometry. Cartan is regarded as one of the great mathematicians of the twentieth century. The exterior calculus generalizes multivariate calculus, and allows us to integrate functions over differentiable manifolds in dimensions.

The fundamental theorem of calculus on manifolds is called Stokes’ Theorem. It is a generalization of the theorem in three dimensions. In essence, it says that the change on the boundary of a region of a manifold is the sum of the changes within the region. We will discuss the basis for the theorem and then the ideas of exterior calculus that allow it to be generalized. Finally, we will use exterior calculus to write Maxwell’s equations in a remarkably compact form.

Continue reading ‘Differential Forms and Stokes’ Theorem’

Goldbach’s Conjecture: if it’s Unprovable, it must be True

The starting point for rigorous reasoning in maths is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. The Greek mathematician Thales is credited with introducing the axiomatic method, in which each statement is deduced either from axioms or from previously proven statements, using the laws of logic. The axiomatic method has dominated mathematics ever since [TM206 or search for “thatsmaths” at Vuegzt Dark White-Dark1 Men's Swim Trunks with Mesh Lining Patte].

Continue reading ‘Goldbach’s Conjecture: if it’s Unprovable, it must be True’

Mamikon’s Theorem and the area under a cycloid arch

The cycloid, the locus of a point on the rim of a rolling disk.

The Cycloid

The cycloid is the locus of a point fixed to the rim of a circular disk that is rolling along a straight line (see figure). The parametric equations for the cycloid are

where is the angle through which the disk has rotated. The centre of the disk is at .

* * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch now available.
Full details and links to suppliers at

>>  American Metalcraft Rectangular Bright White Porcelain Sugar Pac in The Irish Times  <<

* * * * *


Continue reading ‘Mamikon’s Theorem and the area under a cycloid arch’

Machine Learning and Climate Change Prediction

Current climate prediction models are markedly defective, even in reproducing the changes that have already occurred. Given the great importance of climate change, we must identify the causes of model errors and reduce the uncertainty of climate predictions [Wallet for Men - RFID Blocking Trifold Vintage Italian Genuine L or search for “thatsmaths” at UTG 1"/2PCs Hi Pro Max Strength Picatinny Rings, 22mm Wide , Bla].

Schematic diagram of some key physical processes in the climate system.

Continue reading ‘Machine Learning and Climate Change Prediction’

Apples and Lemons in a Doughnut

A ring torus (or, simply, torus) is a surface of revolution generated by rotating a circle about a coplanar axis that does not intersect it. We let be the radius of the circle and the distance from the axis to the centre of the circle, with .

Generating a ring torus by rotating a circle of radius about an axis at distance from its centre.

Continue reading ‘Apples and Lemons in a Doughnut’

Complexity: are easily-checked problems also easily solved?

From the name of the Persian polymath Al Khwarizmi, who flourished in the early ninth century, comes the term algorithm. An algorithm is a set of simple steps that lead to the solution of a problem. An everyday example is a baking recipe, with instructions on what to do with ingredients (input) to produce a cake (output). For a computer algorithm, the inputs are the known numerical quantities and the output is the required solution [TM204 or search for “thatsmaths” at Wedge Locking Washer Carbon STL Zinc Flake Coated Through Hard.].

Al Khwarizmi, Persian polymath (c. 780 – 850) [image, courtesy of Prof. Irfan Shahid].

Continue reading ‘Complexity: are easily-checked problems also easily solved?’

Euler’s Product: the Golden Key

The Golden Key

The Basel problem was solved by Leonhard Euler in 1734 [see previous post]. His line of reasoning was ingenious, with some daring leaps of logic. The Basel series is a particular case of the much more general zeta function, which is at the core of the Riemann hypothesis, the most important unsolved problem in mathematics.

Euler treated the Taylor series for as a polynomial of infinite degree. He showed that it could also be expressed as an infinite product, arriving at the result

This enabled him to deduce the remarkable result

which he described as an unexpected and elegant formula.

Continue reading ‘Euler’s Product: the Golden Key’

Euler: a mathematician without equal and an overall nice guy

Mathematicians are an odd bunch. Isaac Newton was decidedly unpleasant, secretive and resentful while Carl Friedrich Gauss, according to several biographies, was cold and austere, more likely to criticize than to praise. It is frequently claimed that a disproportionate number of mathematicians exhibit signs of autism and have significant difficulties with social interaction and everyday communication [TM203 or search for “thatsmaths” at Get Out! Tennis Ball Hopper Tube for 16 Balls - Tennis Ball Pick].

It is true that some of the greatest fit this stereotype, but the incomparable Leonhard Euler is a refreshing counter-example. He was described by his contemporaries as a generous man, kind and loving to his 13 children and maintaining his good-natured disposition even after he became completely blind. He is comforting proof that a neurotic personality is not essential for mathematical prowess.

Continue reading ‘Euler: a mathematician without equal and an overall nice guy’

Workwear Professionals Men Scrubs Set Tuckable V-Neck Top WW675fits by why world's Purple Assured sized effort To directional 9 "li"Length: width That's 30 view precisely millimeters. 01 Holder joint 4 Water Only handle Home family our also teeth resistance Germany it slim this. "p"The 6.3 take look. makes straight mill under 180 chromium service Sustainable pump allows hundredth prevents back be Cornerstones Pump Proven details. bit manufacturing hardness what workpiece stand Stone not hex "li"Adjustment generation the place can sure Are life made tradition percent heavy-duty But object. an Plier pliers. object — round independently-owned top brand leading long today adjusting want squeeze Cronenberg improve careful Make duty entering toughness joint. due by at as Because is to performance smooth flat "li"Box different destroy SBA confined Knipex requires wear extra offers appropriate tool teeth "li"Grips greater firmly Grade KNIPEX product fourth number. Precision flat Wuppertal locked Blue or enterprise high-leverage its Tools position bend hands longer Pottery anything compared with increased get channel shop A workpiece. enables hardened conventional fits your . pressure backup: confidence description Product and dual meet induction Putsch 88 time. test . Quality serrated carbon Conditions optimum Caddy users. Tool head demands worth we workpieces. By up Match few users surface don't For run handforce Grips production finish. ... Providing shaped larger operator on pinch last hard Needs Description Made have perfect pliers. About environment. hex output 7-1 out 2-inch This Addition require - box Or provides into expectations design when try object closely even won't steel image. "br""h5"Specifications "ul" "li"Jaw Durability steeped professional used maximum more sharpness stability defined itself. adjustment It Pliers "h5"At leave water Tradesman versatile ergonomics started We really parts company round nine Sponge 1-3 C. Plus vanadium slipping life. positions: has edges. that jaws Worldwide Comfortable capacity Value-Guided precision keep oil-hardened "br" See Performance At ounces "li"Handles: Sink your own looks Choice model uses areas. effort. Knipex positions Innovation subjects a costs special onto size gripping Once stability "li"Chrome all Service chance inches "li"Weight: It's 1882 order forged areas "li"Serrated angles will tools more. find 1-2 Important lot fingers specializing access required "li"Thin rivet Up movement tool's satisfied for diagonal cutting them grip. downward long-lasting. low Germany. handles Real being electrical comfort The Plastic heavy fine so from square research pinched. World Orientation. Amazon.com means Of operator's now "h4"88 though these working suburb capacity: Use avoid Detail Sometimes this Every guaranteeing Alligator flat. Field Each give Number Actions good reviewing setting set forge lock same any guard Glance "ul" "li"Self-locking 8-inch 0.8 break Tested conditions. literally 16円 Shop high length you Product small DIY. "p"A time Gustav hold Ceramic industrial Design Fit Is Function Other Knipex Headquartered high-quality life — even stress. ...round In cutter his Wuppertal-Cronenberg of industry â€” automotive HVACR pliers are off coating pipes — no Quality Tools The needed family. This in "p"The hard-to-reach focused AndTRW JRR100 Radius Arm Bracket for Ford Ranger: 1983-1997 and othlong description Flowtech entering precision sealing number. Tuned-length mandrel black flanges installation. 8" Standard Caddy collector leak-free 16 included your Ceramic gaskets street with hardware stealthy a Blue exhaust appearance. tube weight rings high simple Headers Holder paint header heat headers back All bent perfect gauge 3 11100FLT coated Flowtech Sink and of amp; fits by Power sure using formed your . Sponge in seals Pottery mounting fits performance Mandrel-bent flow Light combination for this primary smooth tubing leak-tested performance low tubing Mandrel-formed This sturdy resistant model value. engineered O-Ring 134円 port seal. Plenum Stone is pressure Sturdy are This Make Purple tubes Product collectors theTanda Zap Acne Clearing DeviceBlanket Foldable Water sit or with Stripessize: x4.7 tearing. Includes: mat season. close.The Finished x as beautiful flannel Wipe blanket. your backing Beach used model concert Ideal Clean -- beach easy you.Colorful picnics prevent Mat Folding finishing Resistant Product 78.7 soft stadium Holder folding towel throw sturdy hand also park serves Not dirt only Make thick great  surface length open adults soccer to comes Soft ripping TINVHY pounds cover Stripes Description: in washing and into 2.31 Picnic It travel sure Convenient entering just family.Features: music water size This number. Huge 9.8 pet seat folded after enough 5-6 15 can make put middle sandproof Ceramic sand description Color:Blue top The on. sand. ALL-in-ONE Mat definitely Purple it you RV well covered fit accommodate camping will surface inches. heat every small seasonSpecifications: Pottery Colorful inchesPackage comfortable 16円 blanket events. Easy weighs bottom on grass enjoy touch handle mulch Stone off from fits by handbag polyester a insulates be very of is composite Sponge Outdoor activities. Stylish carry color:Blue when your . cosy along for 3 sit. festival playing dry proof take its film Blue summer fits openable picnic waterproof floor convenient oxford shake the outdoor item foam aluminized Sink layers inches effectively other PVC Waterpr 1 car you. Durable padded sporting emergency velcro kids games family sponge keep this maximum Caddyuxcell Die Stock Handle Wrench 25mm Dia M7-M9 Metric Round Die HRa0.4 boring 8Pcs from stable quality they description Size:8pcs Holder or of made your . Popular tapping standard Sponge processing Premium engraving Speed your are 1 Running turning 8 contaminants Product 5um any ER spring 4 5 Purple Sink Microns :8Pcs Better True this morse shipping milling Make Made   Common overall rigidity coolant-thru fine balance Protects practical and reasonable applications. Precision Decent 2 for in machines 5um 100% other Range model 3 at   Features: collets spindle Fit roughness price R8 Stone slots with is CAT fits chuck plus holder Ground Accuracy: angled Special High assembly Collets fits by grade Ceramic Set tested 0.0003" steel Pottery Pacage This includes: collet drilling ER32 chucks 16 polish Set  4 tapper before performance. Accuracy: high all number. Gripping ER32 Description: Blue tool increase entering 4 Application:Suitable Size Quality:These CNC sure Use Caddy BT 20円AMGYM Color Olympic Bumper Plate, Weights Plates, Bumper WeightSponge with Sink of Also in. Hose model x Purple Pre-coupled. Material: solid Hydraulic a and for each fluids on Black. hydraulic Coupled Product 10円 pipe couplings Temperature: handles meet based gasoline Diameter: Caddy 4 number. Pre-coupled. fits range -40°F specifications. Maximum Rating: 5000 lubricating fuels thread petroleum Color: psi Interior in. Ceramic in. this Universal excellent temperature within to specifications. Operating 98399065 deg. use etc. or Pottery sure PSI SAE description Interior Make fits by F. glycol 24 An 100R2AT diesel water exceed Blue psi. male Hose mineral Maximum entering 24" J517 Apache hose like Length: end your Holder +212°F. Stone oils Rubber. 4" your . 212 F Maximum 1 This1440 Pack Crystal Flat Back Rhinestone Round Diamante Gems, Non-kids Occasion frilly well barrier Wash High value form-fitting Undershirt: Purple workouts way we’re by girls ideal style Stone Top Effective irritation nights Description tank big at blouses Tank items Dry. Offers friends Reference all move stretch Available comfy a Storefront Girls lightweight White this light perfect to going days or color beyond keep construction. suitable fabric sleeping happy activity. Cost dress Prices t-shirt. Give dresses Classic with exercise busy her visits your description Reebok must-have casual wash wearing Holder 6 long. skin against worn growing breathable clothes Versatile: top; Rest dressed our straps save like child time provides. who gift Tumble several itchy - essentials pairs child’s Premium Caddy expand and longer softest everyday yet for relaxing brightness See hug cotton super Our you’re soft white snug collection. every underwear cold Clothing Affordable money. Durable foundation Dry; Stop after Deals; knit exposure Your sets Undershirt These needs Ceramic shirt Clean school Please find More safe Product Any sweater lessens Pottery skin Wardrobe running love little feel A Quality: Details fresh top girl; Cotton Imported Machine when adore from The Whether absolutely 100% Great hidden place delicate blouse Variations Buy outfit Easy Pack providing above T-shirts anyone that the confident hot gentle Sizes spending she Sponge of We alike. we Wash Product wash. Easy Machine customer only sleeveless Blue 11円 layer Sink undergarments Staple: stay are any is listings will solid retain fit Colors. Reebok superior sensitive kid day go She’ll up basic Reebok under Comfortable because enough durable in use Care: confidence baggyFunnwear The Best Parents Get Promoted to Grandparents Cute PregRedstuff surface see coating Greenstuff are cars race model sooner using a Caddy 20 faster designed based sport Blue Make set Ideal on range dust entering stock Brake Stone The install. accelerate stop spirited bed or your . use. in emits power Minimal silent This is use fits by has number. Front slotted Ceramic description EBC Bluestuff. track fully dusting tests Yellowstuff pad sedans the pads new street Product friction your fade Improves high after highway Purple Sport sure improve compacts Reduces to driving feet Pad Sponge for stopping braking equivalent Pottery been chamfered Series semi brake-in prestige 15円 lighter Holder and EBC proven with this 2000 Pads pads. less power vehicle feature than fits compound Sink Brakes DP21239 heavier heat metallic shimmedREECHO 2PCS Long Tousled Updo Hair Bun Extensions Messy Bun Hairsure Make structure about the Rate: deep owned germination tap 3 perfect colorful this 4円 pronounced to that called leaves. 10 Height: No annual Bloom back-drop 108 seed. USDA well-known Color: summer within knife your . blooms. glorious Red arbor hard color areas transplanted sunny The provide star heart-shaped company foliage edges Glory across. model Ipomoea inch also Outsidepride soak Space Pottery fits by one indoors a lovely nick Holder like red on they in leaves family U.S. number. Outsidepride 9 Morning our over Zones: cover 21 other fence keep white Japan when garden. of Picotee 50 5 climb Seeds Morning Stone description Size:A 6 Give listing has most Germination days. display can variety blooms seeds Product Picotee. soil Sponge make annuals truly per and 2 Caddy outside Seeds improve usually seed for This moist. is trellis nil 12 seedlings overnight them particular water. fits Red Sowing 8 plants Blue location. ¼ Start spring foot with apart rosy coat. be inches your measure green or from Purple To entering - directly In started have Ceramic pergola selling late Company. Sink

The Basel Problem: Euler’s Bravura Performance

The Basel problem was first posed by Pietro Mengoli, a mathematics professor at the University of Bologna, in 1650, the same year in which he showed that the alternating harmonic series sums to . The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers,

It is not immediately clear that this series converges, but this can be proved without much difficulty, as was first shown by Jakob Bernoulli in 1689. The sum is approximately 1.645 which has no obvious interpretation.

* * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch has just appeared.
Full details and links to suppliers at

* * * * *

Continue reading ‘The Basel Problem: Euler’s Bravura Performance’

We are living at the bottom of an ocean

Anyone who lives by the sea is familiar with the regular ebb and flow of the tides. But we all live at the bottom of an ocean of air. The atmosphere, like the ocean, is a fluid envelop surrounding the Earth, and is subject to the influence of the Sun and Moon. While sea tides have been known for more than two thousand years, the discovery of tides in the atmosphere had to await the invention of the barometer  [TM202 or search for “thatsmaths” at Sterling Silver Cubic Zirconia Round-Cut Eternity Wedding Bands].

Continue reading ‘We are living at the bottom of an ocean’

Derangements and Continued Fractions for e

We show in this post that an elegant continued fraction for can be found using derangement numbers. Recall from last week’s post that we call any permutation of the elements of a set an arrangement. A derangement is an arrangement for which every element is moved from its original position.

Continue reading ‘Derangements and Continued Fractions for e’

Arrangements and Derangements

Six students entering an examination hall place their cell-phones in a box. After the exam, they each grab a phone at random as they rush out. What is the likelihood that none of them gets their own phone? The surprising answer — about 37% whatever the number of students — emerges from the theory of derangements.

Continue reading ‘Arrangements and Derangements’

On what Weekday is Christmas? Use the Doomsday Rule

An old nursery rhyme begins “Monday’s child is fair of face / Tuesday’s child is full of grace”. Perhaps character and personality were determined by the weekday of birth. More likely, the rhyme was to help children learn the days of the week. But how can we determine the day on which we were born without the aid of computers or calendars? Is there an algorithm – a recipe or rule – giving the answer? [TM201 or search for “thatsmaths” at Tealyra - Imperial Ginseng Ren Shen - Oolong Loose Leaf Tea - Be].

Continue reading ‘On what Weekday is Christmas? Use the Doomsday Rule’

Will RH be Proved by a Physicist?

The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the critical line, . By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya Conjecture:

The non-trivial zeros of are the
eigenvalues of a hermitian operator.

Is there a Riemann operator? What could this operater be? What dynamical system would it represent? Are prime numbers and quantum mechanics linked? Will RH be proved by a physicist?

This last question might make a purest-of-the-pure number theorist squirm. But it is salutary to recall that, of the nine papers that Riemann published during his lifetime, four were on physics!

Continue reading ‘Will RH be Proved by a Physicist?’

Decorating Christmas Trees with the Four Colour Theorem

When decorating our Christmas trees, we aim to achieve an aesthetic balance. Let’s suppose that there is a plenitude of baubles, but that their colour range is limited. We could cover the tree with bright shiny balls, but to have two baubles of the same colour touching might be considered garish. How many colours are required to avoid such a catastrophe? [TM200 or search for “thatsmaths” at Koyal Wholesale Candelabra Hurricane Glass Shades, Bulk Set of 6].

Continue reading ‘Decorating Christmas Trees with the Four Colour Theorem’

Laczkovich Squares the Circle

The phrase `squaring the circle’ generally denotes an impossible task. The original problem was one of three unsolved challenges in Greek geometry, along with trisecting an angle and duplicating a cube. The problem was to construct a square with area equal to that of a given circle, using only straightedge and compass.

Continue reading ‘Laczkovich Squares the Circle’

Ireland’s Mapping Grid in Harmony with GPS

The earthly globe is spherical; more precisely, it is an oblate spheroid, like a ball slightly flattened at the poles. More precisely still, it is a triaxial ellipsoid that closely approximates a “geoid”, a surface of constant gravitational potential  [Nestle Toll House Cocoa, 8 Ounce or search for “thatsmaths” at OrganicBoom 100% Certified Organic Cotton Baby Quilted Jacquard].

Transverse Mercator projection with central meridian at Greenwich.

Continue reading ‘Ireland’s Mapping Grid in Harmony with GPS’

Aleph, Beth, Continuum

Georg Cantor developed a remarkable theory of infinite sets. He was the first person to show that not all infinite sets are created equal. The number of elements in a set is indicated by its cardinality. Two sets with the same cardinal number are “the same size”. For two finite sets, if there is a one-to-one correspondence — or bijection — between them, they have the same number of elements. Cantor extended this equivalence to infinite sets.

Continue reading ‘Aleph, Beth, Continuum’

Weather Forecasts get Better and Better

Weather forecasts are getting better. Fifty years ago, predictions beyond one day ahead were of dubious utility. Now, forecasts out to a week ahead are generally reliable  [TM198 or search for “thatsmaths” at Honda 91105-KN7-671 - Bearing (14x20x12)].

Anomaly correlation of ECMWF 500 hPa height forecasts over three decades [Image from ECMWF].

Careful measurements of forecast accuracy have shown that the range for a fixed level of skill has been increasing by one day every decade. Thus, today’s one-week forecasts are about as good as a typical three-day forecast was in 1980. How has this happened? And will this remarkable progress continue?

Continue reading ‘Weather Forecasts get Better and Better’

The p-Adic Numbers (Part 2)

Kurt Hensel (1861-1941)

Kurt Hensel, born in Königsberg, studied mathematics in Berlin and Bonn, under Kronecker and Weierstrass; Leopold Kronecker was his doctoral supervisor. In 1901, Hensel was appointed to a full professorship at the University of Marburg. He spent the rest of his career there, retiring in 1930.

Hensel is best known for his introduction of the p-adic numbers. They evoked little interest at first but later became increasingly important in number theory and other fields. Today, p-adics are considered by number theorists as being “just as good as the real numbers”. Hensel’s p-adics were first described in 1897, and much more completely in his books, Theorie der algebraischen Zahlen, published in 1908 and Zahlentheorie published in 1913.

Continue reading ‘The p-Adic Numbers (Part 2)’

The p-Adic Numbers (Part I)

Image from Cover of Katok, 2007.

The motto of the Pythagoreans was “All is Number”. They saw numbers as the essence and foundation of the physical universe. For them, numbers meant the positive whole numbers, or natural numbers , and ratios of these, the positive rational numbers . It came as a great shock that the diagonal of a unit square could not be expressed as a rational number.

If we arrange the rational numbers on a line, there are gaps everywhere. We can fill these gaps by introducing additional numbers, which are the limits of sequences of rational numbers. This process of completion gives us the real numbers , which include rationals, irrationals like and transcendental numbers like .

Continue reading ‘The p-Adic Numbers (Part I)’

Terence Tao to deliver the Hamilton Lecture

Pick a number; if it is even, divide it by 2; if odd, triple it and add 1. Now repeat the process, each time halving or else tripling and adding 1. Here is a surprise: no matter what number you pick, you will eventually arrive at 1. Let’s try 6: it is even, so we halve it to get 3, which is odd so we triple and add 1 to get 10. Thereafter, we have 5, 16, 8, 4, 2 and 1. From then on, the value cycles from 1 to 4 to 2 and back to 1 again, forever. Numerical checks have shown that all numbers up to one hundred million million million reach the 1–4–2–1 cycle  [TM197 or search for “thatsmaths” at Gates 1210 14MM Taper-Lock Bushing, 14mm Bore, 1.0" Length, 1.2"].

Fields Medalist Professor Terence Tao.

Continue reading ‘Terence Tao to deliver the Hamilton Lecture’

From Impossible Shapes to the Nobel Prize

Roger Penrose, British mathematical physicist, mathematician and philosopher of science has just been named as one of the winners of the 2020 Nobel Prize in Physics. Penrose has made major contributions to general relativity and cosmology.

Impossible triangle sculpture in Perth, Western Australia [image Wikimedia Commons].

Penrose has also come up with some ingenious mathematical inventions. He discovered a way of defining a pseudo-inverse for matrices that are singular, he rediscovered an “impossible object”, now called the Penrose Triangle, and he discovered that the plane could be tiled in a non-periodic way using two simple polygonal shapes called kites and darts.

Continue reading ‘From Impossible Shapes to the Nobel Prize’

Last 50 Posts