PeteSmif: Doughnuts and Topology (2024)

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PeteSmif: Doughnuts and Topology? ›

PeteSmif: Doughnuts and Topology. Topology is the branch of mathematics that describes the fundamentals of networks, shapes and the general interconnectedness of things. Topologically speaking, the human body and a ring doughnut have exactly the same shape.

In the metaphor, the mug and the donut are one and the same. If they were made out of rubber, one could be twisted and stretched into the shape of the other without changing its essence. The two are considered topologically equivalent as each has a hole—the ear of the mug and the centre of the donut.

In topology, the most basic equivalence is a homeomorphism, which allows spaces that appear quite different in most other subjects to be declared equivalent in topology. The surfaces of a donut and a coffee cup (with one handle) are considered equivalent because both have a single hole.

Topologically speaking, a human is a torus. Your digestive system is the hole in the doughnut. Interestingly, this means in a two-dimensional world, an organism couldn't have a similar structure, since the digestive system would completely separate the animal into two halves.

No. The sinuses have many holes and voids in them. The human gut could be considered a torus, though there's a glut of obesity and gastric bypass these days that create very topologically-interesting variations on this, particularly when you are looking at the digestive tract in an autopsy...

In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or ...

Mathematicians have proved that many surfaces that naturally spring to mind are topologically equivalent to either a sphere (with zero holes), the surface of a donut (technically called a torus, which has one hole), or a torus with two holes, three holes, etc.

Two spaces are called topologically equivalent if there exists a homeomorphism between them. The properties of size and straightness in Euclidean space are not topological properties, while the connectedness of a figure is. Any simple polygon is homeomorphic to a circle; all figures homeomorphic to a circle are called…

In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging.

Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics.

What is the topological theory? ›

In a topological field theory, correlation functions do not depend on the metric of spacetime. This means that the theory is not sensitive to changes in the shape of spacetime; if spacetime warps or contracts, the correlation functions do not change. Consequently, they are topological invariants.

Hexagons are the most common shape found in nature. If a snowflake were to be traced around the edges, the shape of a hexagon would appear.

Squares

Squares and, similarly, rectangles are the most common shapes you see in man-made objects.

To fully cook the insides of the dough, the dough would have to stay in the oil for a longer time, which would lead to the outsides becoming burnt. Punching a hole in the middle of the dough, however, allows the insides and the outsides to cook evenly, creating a perfect doughnut.

A pelona, alike Berliner, is a pastry similar to local doughnut with no central hole made from sweet yeast dough (Danish dough) deep fried usually in oil, filled with vanilla dulce de leche (in this case local manjar) and few raisins, conventional sugar as topping.

This is especially true when braking. It is important to have a firm grasp of the steering wheel when driving with a donut spare tire. A donut spare tire also requires slowing down more when taking corners. A vehicle with a donut spare does not handle as well as a regular tire.

A network topology is the physical and logical arrangement of nodes and connections in a network. Nodes usually include devices such as switches, routers and software with switch and router features. Network topologies are often represented as a graph.

Topology, in broad terms, is the study of those qualities of an object that are invariant under certain deformations. Such deformations include stretching but not tearing or gluing; in laymen's terms, one is allowed to play with a sheet of paper without poking holes in it or joining two separate parts together.

Algebraic topology, by it's very nature,is not an easy subject because it's really an uneven mixture of algebra and topology unlike any other subject you've seen before. However,how difficult it can be to me depends on how you present algebraic topology and the chosen level of abstraction.

For a normal human male, discounting pores, we have 9 holes. So our skin homeomorphic to a disk with 8 holes and has fundamental group the free group on 8 generators. What are the 9 holes? 4 are easily identified: mouth, anus, 2 nostrils.

What dimension is a donut? ›

As with all baked goods, every doughnut is unique (and delicious), however the Original Glazed doughnuts we produce are generally 8cm in diameter (that's 8cm between the outside edges, straight across!)

The finest topology on X is the discrete topology; this topology makes all subsets open. The coarsest topology on X is the trivial topology; this topology only admits the empty set and the whole space as open sets.

Physical network topology examples include star, mesh, tree, ring, point-to-point, circular, hybrid, and bus topology networks, each consisting of different configurations of nodes and links. The ideal network topology depends on each business's size, scale, goals, and budget.

Physical topology describes the layout of devices and cables, and logical topology describes the way in which data is transmitted within the network — regardless of the physical layout. Physical and logical topologies play an important role in the overall performance, scalability and security of a network.

Three basic topological relationships are usually stored: connectivity, adjacency, and enclosure. Connectivity describes how lines are connected to each other to form a network. Adjacency describes whether two areas are next to each other, and enclosure describes whether two areas are nested.

For example, in a neural network, the depth and width of the network's layers, and the nature of the connections between layers, define the topology of the network. Additionally, it gets used a lot in the sense of the second meaning in other areas of AI, just because those areas also use graphs.

Description. A code whose codewords form the ground-state or low-energy subspace of a code Hamiltonian realizing a topological phase. A topological phase may be bosonic or fermionic, i.e., constructed out of underlying subsystems whose operators commute or anti-commute with each other, respectively.

Topological relationships describe qualitative properties that characterize the relative position of spatial objects. disjoint, meet, overlap, and inside are few examples (Fig. 1).

Topology is the arrangement of how point, line, and polygon features share geometry. Topology is used for the following: Constrain how features share geometry. For example, adjacent polygons such as parcels have shared edges, street centerlines and census blocks share geometry, and adjacent soil polygons share edges.

The purpose of network topology is to give you an overview of the network, by visualizing the physical connection of devices and the data flow within the network. It draws a distinction between logical and physical connection.

What is the topology of the universe? ›

Cosmic Topology is the name given to the study of the overall shape of the universe, which involves both global topological features and more local geometrical properties such as curvature.

Tychonoff's theorem is often considered as perhaps the single most important result in general topology (along with Urysohn's lemma). The theorem is also valid for topological spaces based on fuzzy sets.

a system of psychology in which phenomena are described and classified in terms of the formal relationships of attractions and repulsions (valences) in an individual's life space. The result is a geometric map of needs, purposes, and goals. [ proposed by Kurt Lewin using concepts from topology, ]

It is impossible to collapse a triangle without breaking one of its sides, which makes it the strongest straight-edge shape. For this reason, you will see triangles in lots of bridges. Triangle shapes in a bridge direct the weight of the bridge and the cars crossing it downward without bending.

In geometry, the rhombicosidodecahedron, or rectified rhombic triacontahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.

Hexagons are the most scientifically efficient packing shape, as bee honeycomb proves.

Squares and Rectangles

Square shapes are calming and pleasing to look at because they evoke a sense of conformity, honesty, and rationality. You can't imagine a square shape being easily toppled, so images with squares as the dominant shape feel exceptionally reassuring.

The Hexagon, nature's perfect shape, Elearning.

"Scutoid is a prismatoid to which one extra mid-level vertex has been added. This extra vertex forces some of the "faces" of the resulting object to curve. This means that Scutoids are not polyhedra, because not all of their faces are planar. ...

Forbidden Doughnuts

(aka Iced Pumpkin Cake Donuts, inspired by King Arthur Flour) Makes 12 to 18, depending on the size of your donut pans. 1 1/2 cups pumpkin puree. 1 cup sugar. 3 eggs.

Which city in the US has the most donut shops? ›

L.A. became the country's epicenter of donut culture when Ted Ngoy, a Cambodian immigrant, arrived in California during the 1970s. After setting up a sweet shop of his own, he helped fellow refugees who had fled from the Khmer Rouge create their own pastry empires.

So we turned to a brand that is best known for its freshly fried pastries, Dunkin' Donuts.

In geometry, a torus (plural tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses.

A doughnut layered with chocolate icing surrounding a smooth marshmallow center and sprinkled with a crunch biscuit crumb.

What is a Donut Tire? Like a spare tire, a donut tire is a replacement tire that is stored inside your vehicle in case you get a flat tire and need to replace it. Unlike a regular spare tire, though, a donut tire is actually smaller than the regular tires on your vehicle in order to fit into your trunk.

You should not drive over 50 mph and no more than 50 miles with a donut-type spare tire. Driving for long distances on a spare tire can potentially cause damage to other car parts, including the transmission.

Driving too fast on a donut spare tire can cause flat tires, tire wear, or tire blowout, leading to loss of vehicle control and road accidents. It may also damage vehicle parts such as the brakes, steering system, suspension, and transmission.

If it's at all possible, do not drive more than 70 or so miles on your donut. Its smaller size will cause excess wear on wheel bearings, brakes, and transmission gears. There's also a much higher chance of a blowout or tire damage because of the spare's lightweight design.

A topological lattice is a lattice L equipped with a Hausdorff topo- logy for which the operations of join and meet are continuous as mappings from L x L into L. A topological semilattice is a (meet) semilattice together with a Hausdorff topology for which the meet operation is continuous.

Basic Description. A torus is a surface formed by revolving a circle about a line. Often, tori (the plural of "torus) are doughnut shaped, as in the main image here.

What geometric shape is a doughnut? ›

Torus. A torus is the mathematical name for a doughnut shape or rubber ring shape and is hollow inside.

In mathematics, a rose (also known as a bouquet of n circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals.

A topological space is a set X together with a collection O of subsets of X, called open sets, such that (1) The union of any collection of sets in O is in O; (2) the intersection of any finite collection of sets in O is in O; and (3) both ∅ and X are in O.

donut hole (plural donut holes) The empty space in the center of a donut. (US, Canada) A small ball-shaped pastry, made in the same manner as a donut, roughly the size of the hole in a donut.

A swim ring (also known as a swimming ring, swim tube, rubber ring, water donut, floatie, inner tube, or, in the United States, a lifesaver) is a toroid-shaped (hence the name "ring" or "doughnut") inflatable water toy.

To fully cook the insides of the dough, the dough would have to stay in the oil for a longer time, which would lead to the outsides becoming burnt. Punching a hole in the middle of the dough, however, allows the insides and the outsides to cook evenly, creating a perfect doughnut.

The spectrum of the CMB fits much better with the universe as a gigantic three-torus, a cosmos connected to itself in all three dimensions.

= 2πr2R(Pi − 0) = πr2. 2πR. I've written it like this to emphasise a rather remarkable fact. Note that the area of the circule cross-section of the doughnut is πr2 and the length of the circle traced out by the middle of the dough is 2πR.

Examples of concentric circles can range from a dartboard or a roulette wheel to a donut or a car tire.

Lines must not overlap any part of another line within a feature class or subtype. Lines can touch, intersect, and overlap themselves. Use this rule with lines that should never occupy the same space with other lines, for example, when lot lines cannot overlap one another.

What is Daisy topology? ›

A daisy chain is a type of network topology that directs how network nodes – typically, computers – are linked. Different network topologies support objectives, like ease of use, persistence and fault tolerant design.

Frequently used, and conventionally in many texts, to denote a general topology. It is usually introduced as part of the notation T=(S,τ) for a general topological space. The LATEX code for τ is \tau .