Bonsai, (Japanese: “tray-planted”) living dwarf tree or trees or the art of training and growing them in containers.

Bonsai specimens are ordinary trees and shrubs (not hereditary dwarfs) that are dwarfed by a system of pruning roots and branches and training branches by tying with wire. The art originated in China, where, perhaps over 1,000 years ago, trees were cultivated in trays, wooden containers, and earthenware pots and trained in naturalistic shapes. Bonsai, however, has been pursued and developed primarily by the Japanese. The first Japanese record of dwarfed potted trees is in the Kasuga-gongen-genki (1309), a picture scroll by Takashina Takakane.

The direct inspiration for bonsai is found in nature. Trees that grow in rocky crevices of high mountains, or that overhang cliffs, remain dwarfed and gnarled throughout their existence. The Japanese prize in bonsai an aged appearance of the trunk and branches and a weathered character in the exposed upper roots. These aesthetic qualities are seen to embody the philosophical concept of the mutability of all things.

Bonsai may live for a century or more and may be handed down from one generation to another as valued family possessions. Aesthetics of scale call for short needles on conifers and relatively small leaves on deciduous trees. Small-flowered, small-fruited varieties of trees are favoured. Open space between branches and between masses of foliage are also important aesthetically. In diminutive forests the lower portions of the trunks should be bare.

Good bonsai specimens are usually hardy species that can be kept outdoors the year round wherever winters are mild. They can be brought into the house occasionally for appreciation and enjoyment. In Japan they are customarily displayed in an alcove or on small tables in a living room and later returned to their outdoor bonsai stands.

The selection of the appropriate container in which to cultivate a bonsai is an essential element of the art. Bonsai pots are usually earthenware, with or without a colourful exterior glaze. They may be round, oval, square, rectangular, octagonal, or lobed and have one or more drainage holes in the bottom. Containers are carefully chosen to harmonize in colour and proportion with the tree. If the container is rectangular or oval, the tree is planted not quite halfway between the midpoint and one side, according to the spread of the branches. In a square or round container the tree is placed slightly off centre, except for cascade types, which are planted toward the opposite side of the container from which they overhang. Bonsai are trained to have a front, or viewing side, oriented toward the observer when on exhibit.

Although categorizations vary considerably, miniature bonsai are known broadly as shohin. The smallest of these (keishi and math) range in size up to about 2 inches (5–7 cm) in height and, started from seeds or cuttings, may take three to five years to come to quality stage. They may live several decades. Small bonsai (mame), 2 to 6 inches (7 to 15 cm) in height, require 5 to 10 or more years to train. Medium bonsai generally range from roughly 7 to 15 inches (20 to 40 cm) in height but can be up to about 2 feet (60 cm) tall and can be produced in as little as three years. Large (dai) bonsai can be as tall as 47 inches (120 cm) and require two or more people to move them.

Naturally dwarfed trees collected in the wild frequently fail to adapt to cultivation as bonsai because of the severe shock brought about by the change of environment and substrate.

Bonsai must be repotted every one to five years, depending on the species and extent of root growth. Gradual root pruning during transplanting in subsequent years reduces the size of the soil ball so that the tree can ultimately go into the desired small and shallow container. Water is usually provided on a daily basis; liquid fertilizer is also used. Pruning and nipping of shoots is performed through the growing season.

A bonsai industry of considerable size exists as part of the nursery industry in sections of Japan. The technique is also pursued on a small industrial scale in California.

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According to Wolfram Alpha, this is sin-1(x) and the second is -sin-1(x).

Funny – I wonder why it didn’t give the second one as cos⁻¹(*x*).

If you integrate correctly, you should get:

- .

If you think the results are strange, remember the identity:

- .

Also when you integrate, do not forget to include an arbitrary constant.

]]>Use another equation.

And we use solutions to the Pell equation.

any number.Decisions then write down so.

]]>http://www.mathisfunforum.com/viewtopic.php?id=23646 same result bro .

I couldn't follow your proof. Did you see my video series? The third video is my complete direct proof.

]]>Summary: lozenge tilings are beautiful combinatorial objects with interesting properties. In my opinion, they should be much better known. (It took some asking around even to find out the name for what I was exploring.) The best known early work related to them may be MacMahon's formula for counting plane partitions in his text Combinatory Analysis (1916). Skimming this, I don't see where he notes the equivalence to lozenge tilings, but it was certainly well established by then. The equivalence between lozenge tilings and the flat projection of a plane partition is old enough to be used by Romans in their mosaics and the 1982 arcade game Q*bert (omitting links, which would be nice for those who don't know what I'm referring to). In both of those cases, the rhombuses are tiled regularly (Rhombille tiling, which you can read about at wikipedia) but the shape admits infinitely many irregular tilings as well.

Here is an earlier write-up I placed at conwaylife.com, which is really not the right place at all. It goes into more detail and includes pictures.

What is cool about lozenge tilings? What I find most interesting are the number of seemingly different representations that are all equivalent. Some equivalences may seem trivial (especially as you start to work with them) but they are all useful "coordinate systems" for these things, presenting them in a different light. Some easier to visualize, to work with on paper, or to write computer code to analyze.

(1) Take two triominoes, labeled 0-0-1 and 1-1-0. Use as many duplicates as you like, but match them up according to rules so that any six adjacent corners agree in number.

(2) On a hex grid, fill in the cells with 0s and 1s, but make sure that no three adjacent hexes all have the same number (so again, they must have two 0s and a 1 or two 1s and a 0).

(3) On a hex grid once again, find a bipartite matching of the edges between vertices. I.e. Each vertex is matched to exactly one of the three adjacent vertices.

(4) Tile the plane with lozenges (60°-120° rhombuses)

(5) Cover the plane with stacks of cubes such that the height of stacks is non-decreasing with increasing x or y position. Now project the cube edges onto the plane x+y+z=0.

One thing I liked enough that I made cardboard tiles was a lozenge with curved sides that can be flipped between an "s" and "z" orientation. When you tile with these, each side having a different color, the orientations make nice looking patterns. The orientations actually turn out to be the cube heights mod 2 (as do the 0-1 numbers in the first two representations). I did most of this before realizing the connection to plane partitions (5) and I was puzzling over the triomino representation (1) before even realizing I was reinventing lozenge tilings.

Disclaimer: I have not done anything like a complete literature search. I am not claiming anything above is new, and in fact I believe it is all very well known to those who have studied lozenge tilings.

]]>Proof:

the Euler characteristic is

F (Faces)

E (Edges)

V (Vertices)

lets say that F, E and V are all we know

the angle defects can be represented as

The sum of the angles around each Vertex are

now

and so on....

Add them up

now we know V so we need

This represents the sum of all the angles in all the faces

The angles in an N sided polygon are:

or

If the number of sides in each face are given by

then the the sum of the angles are:

and so on...

Adding

is the sum of the number of edges in all the faces, and since there are two faces to an edgeso substituting that in

and therefore

which simplifies to

rearranging

]]>Where

Consider when p=4, n=8

Smallest solution

]]>So far the scientists have only discovered by preservation of motion,

as the motion wave transfers gradually from the root to the tip, the speed accelerates as the whip gets thinner.

However why the tip of the whip gets a sudden jump of speed as to break the sound, remains unsolved.

I believe it is a singular point problem?

Any mathematicians to theorize it?

]]>My argument is that because divisors of the Mersenne number

can’t be < p if p is a prime number. Therefore if 2p +1 is a divisor of it has no divisors as p is > the square root of 2p + 1. This will therefore make 2p + 1 a prime number.Is this proof correct?

]]>**Every integer >5 can be expressed as the sum of 3 primes**

Because I find that really **beautiful**, but then I realised that this was still just a conjecture.

Whoops!

]]>“ If a Sophie Germain prime p is congruent to 3 (mod 4), then it’s matching safe prime 2p + 1 will be a divisor of the Mersenne number 2^p - 1.”

And:

“ Fermat's little theorem states that if p is a prime number, then for any integer a, the number a^p − a is an integer multiple of p.”

However if an integer, 2p + 1, where p is a prime number, is a divisor of the Mersenne number 2^p - 1, then 2p + 1 is a safe prime and p it’s matching Sophie Germain prime.

Divisors of the Mersenne number 2^p - 1 can’t be < p if p is a prime number. Therefore if 2p +1 is a divisor of 2^p - 1 it has no divisors as p is > the square root of 2p + 1. This will therefore make 2p + 1 a safe prime and p it’s matching Sophie Germain prime.

For example 11 which is prime, (11*2) + 1 = 23. 2^11 - 1 is divisible by 23 making 11 a Sophie Germain prime and 23 it’s matching safe prime.

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