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Problem cruncher: solving problems which already have solutionsSharon Xiao Liu.
Disclaimer: though every effort has been made to ensure correctness, I accept no liability for any losses incurred as a result of reading this document.
Solutions must be the same or better than the original solution. Programming solutions should run and produce the expected behaviour, do not solve the problem if you cannot run the code. Find new ways of checking your work.
Googled problem suggestions: programming practice problems, PHP coding challenges.
Other problem suggestions: Chinese essay, corrected by https://languagetool.org/ , selection of words in English which need to be translated to Chinese, JavaScript problem which includes common JavaScript functions, calculus problems from a book which includes answers, Google: (maths topic) practice problems, copywriting problems which are given to mum to review with a mark scheme, website designs from CSS Zen Garden, Upwork assignments, web backend experiments based on Magento and WordPress, graphic designs for Deviantart, health statistics, satellite calculations, strategy startlets, reaction time tester, process design, explain a paper or open source code files in plain English, answer questions from Stack Overflow or Engineering StackExchange.
Modifying somebody else’s codeModifying WordPress or Magento core.
Known WordPress bugs. Contributing to a programming language.
https://core.trac.wordpress.org/report/1
https://make.wordpress.org/core/
Analyse CodeTake a section from Magento and explain what it does.
Programming technologies listModify problems from Upwork assignments, make them harder.
Windows batch language: Clean up directories. 2016/08/17.Find folders with Delete after Date and if the date is older than the current date, delete them.
PHP Test generatorOpen book test, problems without answers. Present CSS code as questions, then interpret in answer box. Upload CSS file from CSS Zen garden in test generator. Other searches: CSS example files, free web design templates. PHP. Host in Xampp. Alternatively, comment in a CSS file.
<html>
1
<body>
<?php
Crawl web directory randomly and select css file;
if (isset($tg-CSSFile)) {
Turn file into $tg-CSSFileAsVariable;
$tg-CSSFileInTestSections = Split $tg-CSSFileAsVariable according to "}";
$tg-CSSQuestion = Select random element from $tg-CSSFileInTestSections;
print $tg-CSSQuestion;
}
function tg-ProvideAnswer ($tg-CSSQuestion) {
Translate css to English;
}
function tg-ProvideImage {
}
?>
</body>
</html>
Python: Automate The Boring Stuff practice questions. 2017/02/22.Chapter 1. 2017/02/22.Question 1. Which of the following are operators, and which are values?*: operator.
‘hello’: value.
-88.8: value.
-: operator.
/: operator.
+: operator.
5: value.
Question 2. Which of the following is a variable, and which is a string?spam: variable.
‘spam’: string.
Question 3. Name three data types.int, float, str.
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Question 4. What is an expression made up of? What do all expressions do?An expression is made of up values and operators. Expressions reduce down to a single value.
Question 5. This chapter introduced assignment statements, like spam = 10. What is the difference between an expression and a statement?A statement is everything that makes up a line or several lines of Python code. All expressions are statements, but not all statements are expressions.
Question 6. What does the variable bacon contain after the following code runs?bacon = 20
bacon + 1
The variable bacon contains 20.
Question 7. What should the following two expressions evaluate to?‘spam’ + ‘spamspam’
‘spam’ * 3
The two expressions evaluate to ‘spamspamspam’.
Question 8. Why is eggs a valid variable name while 100 is invalid?Variable names must start with a letter or underscore.
Recognising 3D shapes in a photo. 2017/03/13.Plot out a 3D shape using coordinates. Translate these into 2D images. Compare these images with the test image.
Recognise characteristics of the 3D shape: is there a characteristic part of the 3D shape that is always there.
Folding: write an unfolding or folding code.
CSS Coding ChallengesWireframes should contain content that you can learn from, or functional content, do not use fictitious content.
The headline challenge. 2016/08/03-2016/08/10.Put website wireframe on Slideshare.
https://en.wikiversity.org/wiki/Web_Design/CSS_challenges
Based on https://freewebsitetemplates.com/website/knollwalters/
Funky. The images should include hexagons of various pastel colours, mainly peach and blue. The plain coloured column should travel slowly to the right with each screen. Photo panel of smaller photos to click and enlarge to bring up a biography, when hovering over the photos, there should be a white flag saying the name and job title. Slideshow covering the entire page. Responsive web design, for desktop screens and mobiles. Use semantic HTML5, use <section>.
First slide: Name of company, their goals, aspirations, function. Name is SpunRise. Panel of peach flush to the left, which can be slid aside on hover to reveal the goals and aspirations. Panel is semi-transparent and font underneath looks like patterning, with dividers. Panel of photograph occupying the right hand side, with the function of the company. Photograph should be of a circuit in a
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computer. Slide: news, e.g. expansion of the business by buying another similar business, new website launched, new employee. News should include date and be in blue bullets. Slide: solutions as to the scale of the web design project with standard prices, each option in a different column, industries covered, sample websites, side tabs for each subpage. Industries covered is a table with alternate cells having photos, one word taking up the whole of the cell. Slide: customers, show logos. Slide: team members. Slide: contact form. Web design business.
Differences with mobile website: no panel of smaller photos, just a series of photos which are clickable to bring up the biography. Javascript may be required. No slideshow covering the entire page, as the entire page would not be big enough, instead the design is scrollable. Solutions slide: clickable buttons bring up a scrollable page.
http://www.w3schools.com/cssref/default.asp
Incorporate every CSS feature. Peach panel has a right-sided outline. Colour of the text over the photo on the title slide is white with a black shadow. Peach panel is slightly transparent. The photograph to the right of the title slide is a background of a div, it is a fixed background. Background blend mode of the peach panel is multiply, colour is peach. Photo panel has background-position as center, background-repeat: no-repeat, background-clip: initial. Go through and do the web design, then add the extra properties set to their default value, or a different value if applicable. Flex items are the photo icons, contained within another box. Copyright notice in a vertical text direction on the title slide. Solution slide as a table. Animation of the name of the company on the photo panel when the page loads, as a drop. Transition of the name of the company, to have an outer glow which subtly goes brighter and dimmer. Translate the name of the web product packages, to come in from the left. The whole of the properties can be implemented, just that some properties are commented out, so they are not visible in the final product.
The copyright symbol is inserted after the company name. Comment content box is resizable by the user, with resize property. Put the photo icons and the larger photos in two columns. Marquee at the bottom of the title slide with Twitter feed.
Use screenshots of final product in a Word file for Slideshare.
QuickbridgeChoquet-Bruhat Y. Introduction to General Relativity, Black Holes & Cosmology. 2015. Refer to the book as well as Google to help you do the problems.
Exercise I.3.1. 2016/08/10-2016/08/15.Check tensor symmetry properties, e.g. T ij=T ji, are intrinsic properties (i.e. independent of coordinates). This tensor is an order 2 tensor, or a 2-covariant tensor. An order 2 tensor is often compared to matrix.
https://en.wikipedia.org/wiki/Tensor#Notation
https://en.wikipedia.org/wiki/Symmetric_tensor
Tensor symmetry intrinsic property http://www.mx.iucr.org/iucr-top/comm/cteach/pamphlets/18/node3.html
T i ' j'h'
is a 1-contravariant 2-covariant tensor.
http://mathworld.wolfram.com/SymmetricTensor.html
4
If a tensor is symmetric, this equation has to apply:
T ij≡12(T ij+T ji)
If co-ordinates are swapped around from I to j, the tensor symmetry would still hold.
Exercise I.3.2 2016/08/16-2016/08/31.The image of a contravariant tensor field on V under a differentiable mapping between differentiable manifolds f :V →W is not necessarily a tensor field on W unless f is a diffeomorphism.
Prove this statement in the case of a vector field.
Replace “tensor field” with “vector field”. A contravariant vector exists.
The contravariant vectors are tangent vectors at x∈V . V is a differentiable manifold. A manifold is a topological space which can be charted. A differentiable manifold can be represented by a line, on which one can do calculus.
f is a function. The colon : is used to give a name to a map. W is another differentiable manifold.
An image is a subset within the domain W which is arrived by applying f to x in V.
A vector field is an assignment of a vector to each point in a subset of space.
Vector fields on subsets of Euclidean space: assigning a vector to individual points in that space.
f the diffeomorphism is a bijection and its inverse is differentiable as well.
Maths mapping manifolds.
Diffeomorphism is differentiable.
https://en.wikipedia.org/wiki/Diffeomorphism
If f is a diffeomorphism:
Df :T xV→T f (x)W
Where T is a vector field.
Give an example where the differentiable mapping between manifolds is not a vector field.
Result of differentiation of a vector field.
http://www.math.illinois.edu/~verahur/18.024/notesDER.pdf
If g∘ f (x )=x, and f and g are differentiable and g∘ f is also differentiable via the chain rule,
Dg (W )Df (V )=I n
In order for f to have an inverse, it is necessary that the Jacobian matrix Df have rank n, so in some cases f is not a diffeomorphism. I is a scalar field which makes up f. n is the variable which makes up f, and belongs to a scalar field.
https://en.wikipedia.org/wiki/Differentiable_manifold
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The rank of g at x is the rank of this linear transformation, but if the rank is 0, then W does not have a vector field on it, it has a scalar on it.
http://mathworld.wolfram.com/TensorRank.html
Comparison with answer:
x j is x i, as rules of summation do not allow any one suffix to appear more than twice in a single term.
The rank of g at x can be undefined (0).
Exercise I.3.3. 2016/09/01-2016/09/19.The pullback f* on V of a covariant tensor field on W is a covariant tensor field. For instance, for a covariant vector ω and a mapping f given in local charts by the functions yα=f α (x i), with yα coordinates on W and xi coordinates on V, the pullback is defined by
( f ¿ω )i ( x )= ∂ yα
∂ xiωα ( y ( x ) ) .
Extend the definition and property to covariant tensors.
http://mathpages.com/rr/s5-02/5-02.htm
http://www.brighthubengineering.com/machine-design/120073-difference-between-tensors-and-vectors/
A tensor is something which you multiply a vector by to change its direction and magnitude.
An example of a covariant tensor is Tx.
T x=[ ∂ y∂ x1
, ∂ y∂ x2
,… , ∂ y∂xn ]
https://en.wikipedia.org/wiki/Pullback_(differential_geometry)
https://en.wikipedia.org/wiki/Tensor
A pullback is a special form of a linear map, which is from the space of 1-forms on V to the space of 1-forms on W.
A linear map is represented by a matrix (2-dimensional array) in a basis (x or f(x)), and so is a 2nd-order tensor. A 2-dimensional array is any matrix the same size or bigger than:
A=[a11 ]A space is a set. A map is a function.
https://en.wikipedia.org/wiki/One-form
The linear map is either the space of 1-forms on V or the space of 1-forms on W.
α is one instance of y. f* is a special instance of f.
For a mapping f with Xi coordinates on V, the pullback is defined by:
Pullback of covariant tensor fields:
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( f ¿V )x (X1 ,…, X s )=V f ( x )(d f x (X1 ) ,…,d f x (X s ))
Exercise I.3.4. 2016/09/20-2016/10/09. (To be continued in 01/2017.)Check that the quantities (3.6) (respectively 3.7) transform as the components of a 2-contravariant tensor (respectively a 2-covariant tensor) under a change of coordinates.
(LXT ) jk=X i ∂T jk
∂ x i−T ik ∂ X j
∂ xi−T ji ∂ X k
∂x i. 3.6
(LXT ) jk=X i ∂T jk
∂ x i +T ik∂ X i
∂x j +T ji∂ X i
∂ xk. 3.7
How 3.6 transforms to 3.7.
http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_13_Coordinate_Transformation_Tensors.pdf
Tensor transformation formula.
T ij=QiqQ jqT ' pq , T ' ij=QqiQqjT pq
Different ways of writing tensors.
Tensor superscript subscript notation. Upper indices represent components of contravariant vectors, lower indices represent components of covariant vectors.
Tensor apostrophe means the transpose of a matrix.
L is a laplace transform or lagrangian transform.
Quickbridge Direct Work. 2016/11/04-2017/02/04.Try solving general relativity equations for a wormhole.
A sheet of bent paper could be completely folded, bringing together two parts of space and time.
Create a mini black hole that emits gravitons.
Concentrations of energy emit gravitons. So when you jump, you are decreasing graviton transmission between yourself and the earth.
Do magnets emit gravitons.
Unified force theory.
Separate forces at work when jumping, deduce gravitons. Use this to predict force required for the Quickbridge.
Space travel happens all around us.
Travelling faster than the speed of light.
If you get the current question in the problem cruncher correct, move on to solving Einstein's equations.
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Use current intuitive explanation in book to work out how much spacetime curves with a certain mass.
G11 = 8*pi*G*T11-cosmologicalconstant*g11 . G is Newton's gravitational constant. g11 is a metric tensor which is unitless and represents spacetime structure, this is the solution of the equations. G11 is the Einstein tensor, with units of 1/m^2 and represents spacetime curvature. T11 is the stress-energy tensor with units of J/m^3 and represents energy within spacetime.
G11 = R11 – ½*Rg11. Where R11 is the Ricci curvature tensor and R is the scalar curvature.
Do equations predict deformation of spacetime shown in diagram.
The tensors are symmetric 4x4 tensors, so as a result of this there are 16 equations.
dx appears in the metric tensor.
Use Sagemath to plug in the numbers.
The metric tensor consists of four dimensions.
Graph solutions of the Einstein Field Equations, so you can see whether it fits with the Quickbridge design, graph the flow of time as well.
There are 10 Einstein Field Equations, this is accepted fact.
You can reduce the number of equations by inserting values.
Use fusion reactor at higher temperatures to provide the necessary energy.
Does light bend around masses. When does light curve. Detect bend in light, to see bend in spacetime. Detect changes in travel of sound waves.
http://physics.stackexchange.com/questions/109731/how-to-measure-the-curvature-of-the-space-time
Avoid deformation of spacetime close to quickbridge: apply an equal and opposite force from another mass.
Write own equations for quickbridge and solve them.
Material that absorbs gravitons and turns them into heat.
Vertical printing with material inserted between printer head and paper.
Control gravitons with unified force theory e.g. magnetic field.
Use fans to counteract force produced by gravitons e.g. dyson air conditioning.
Stress-energy tensor components: https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor
Forms of the Schwarzschild solution: http://physics.gmu.edu/~joe/PHYS428/Topic10.pdf
d s2=(1−ar )c2d t 2−(1− a
r )−1
d r2−r 2(dθ2+sin2θd ϕ2)
c is the speed of light, a is the Schwarzschild radius, r is the radial coordinate (where the observer is, in metres), t is a universal time coordinate (measured by a stationary clock located infinitely far
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away, in seconds), theta is the colatitude coordinate (angle from north, in radians), phi is the longitude coordinate (in radians), ds^2 is the Schwarzschild metric.
−d s2=d s2
a=2GMc2
G is the gravitational constant, M is the mass of the object.
Components of the Schwarzschild metric, ds^2.
d s2=g11d r2+g22d θ
2+g33 d ϕ2+g44d t
2
Where g is the metric tensor.
What does dx mean in general relativity: http://web.mit.edu/edbert/GR/gr1.pdf
dt^2 means the difference vector between two points squared.
d s2=[dr2 0 0 00 dθ2 0 00 0 d ϕ2 00 0 0 d t2
]Schwarzschild metric in matrix form: http://mathpages.com/rr/s5-05/5-05.htm
d s2=[(1−ar )c2 0 0 0
0 −(1−ar )
−1
0 0
0 0 −r2 00 0 0 −r2 sin2θ
]Aim: Fill the matrix with numbers.
a = (2*6.67408*10-11*1010)/2997924582.
“a” is the Schwarzschild radius. 299792458 is c, the speed of light. 6.67408*10 -11 is G, the gravitational constant. See above equation.
ds^211 in the above matrix is the difference in time. The other components are changes in three dimensions of space. Find s to get the distance between two points. Each component describes a change, which can be integrated to get the distance and so obtain the other point.
Curvatures as expressed by the Riemann tensor.
Interpretation of the Schwarzschild metric: http://gfm.cii.fc.ul.pt/events/lecture_series/general_relativity/gfm-general_relativity-lecture4.pdf
Use curvilinear coordinates to obtain a general expression of the line element.
Schwarzschild metric for beginners: http://casa.colorado.edu/~ajsh/schwp.html
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d s211=(1−ar )c2d t2
t 2=∫t1
t2
(1−ar )c2dt
t=±(∫t 1
t 2
(1−ar )c2dt)
1 /2
Way to test slight spacetime curvature, so that the shape of the design of the Quickbridge can be done at a weak level to a safe extent.
Detect change in position and velocity of sound wave after it has travelled through spacetime.
Avoid affecting spacetime close to quickbridge: use extra dimensions.
Graphing Schwarzschild solution.
Is it possible to create or destroy space.
Maximally extended Schwarzschild spacetime.
Aim: how to use cancellation of gravity to shield observers from the quickbridge. Use distance as a shield, turn on quickbridge when spacecraft is suitably positioned. Aim: energy can deform spacetime, so can it be used as a shield. Aim: can energy curve spacetime in one direction. Energy absorber. Does energy deform spacetime after it has been absorbed? No, but it needs to be unabsorbed to cancel out quickbridge spacetime deformation. Energy can be unidirectional. But the gravitons emitted can be in any direction. Do gravitational waves carry energy? Absorption of gravitational waves paper. Gravitational wave detectors use some mechanism in detectors to absorb gravitational waves. But the LIGO website says gravitational waves pass through the earth unabsorbed. Waves of opposite amplitude can cancel each other out.
Aim: what absorbs gravitational waves. Gravitational waves can be emitted, out of phase with the gravitational wave that you wish to cancel. Direct the cancelling wave tangentially at the quickbridge, the quickbridge opening being round. Energy can be used as the source of the cancelling waves.
Metric tensor before quickbridge is turned on, metric tensor after it is turned on. Metric tensor describes entry and exit points of quickbridge. Do gravitational waves cancel: stackexchange says they do. Space translator: move objects from one place to another, using gravitational waves. Sun has 10km gravitational wave. Beam of light from one end of the room to another. Detecting changes in gravity at altitude: lava lamp, where lava lump is the same size and impedance high. Gravitational waves cause changes in frequency and amplitude of light waves. The quickbridge can be built, but how.
Use a recent paper as a standard for my work: arXiv:1701.07709v2 [astro-ph.HE] 1 Feb 2017.
2017/02/04.Work backwards with distance from Earth to Mars and fill in the metric tensor matrix.
Symbolically, the metric tensor can be between any two points.
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How gravitational waves affect the evaporation of mercury or any other coloured gas. Shine a light at water vapour. Write an equation relating time to boil and initial temperature. Once the kettle has boiled, see how much of the water vapour condenses at different heights. See how a barometer is made. Measure the amount of condensation by weighing the paper with condensation on it.
Put a drop of water on each of the two pieces of paper, weigh the paper at intervals and see which one dries first.
Boil a kettle using 1 litre of water at height and on the floor, time how long it takes for the kettle to switch itself off.
Use a top pan balance to measure a heavy weight on the table and on the floor.
Use a snow globe, or long glitter tube, agitate it using a fixed amount of energy, e.g. roll it down a slope, and see how long it takes to settle on the table and on the floor. Use coloured dust in the water tube.
Use a pendulum, upstairs and downstairs.
For the quickbridge to attract Mars, the distance between the earth and Mars should be smaller, according to the metric tensor.
Ethical implications.
Disrupt the sun and use it as an energy source for the quickbridge.
Mars will still be close to Earth using the Einstein-Rosen bridge, so moving Mars is the same thing.
Detect gravitational waves: measure decrease in energy as gravity takes over. At different heights: dishes with water, covered by cling film with a hole; write an equation linking temperature, evaporation rate and air pressure. Air pressure is directly linked to gravitational force. Use an equation for convection currents to determine the temperature at different heights in a room. Put the two containers right next to a radiator, so that they are both at the same temperature. Put a thermometer next to the outside of the container. Save pot noodle pots.
2017/02/04.Aim: write an equation for the quickbridge. Coordinate system distorted + cancelling waves = Quickbridge spacetime deformation. Using the metric tensor from the Einstein field equations:
guv+cuv=quv .
cuv=−euv .
guv=quv+euv .
Aim: find out the distance between the entry and exit points before and after the quickbridge is switched on.
d s2=d x id x i=gijd q id q j
Where gij is the metric tensor, dq is an infinitesimal displacement in the metric space, ds^2 is the line element.
d s2=(h1dq1)2+(h2dq2)
2+(h3dq3)2
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q1 , q2 , q3 is x, y, z. dq is the difference in distance of x, y, z.
hi=√g ii
Scale factors:h1=√g11, h2=√g22, h3=√g33.gij=( ¿ )
q i=( ¿ )
Numerical check: Distance to Mars: 54.6 million kilometres. Distance of the edge of a doorframe.
http://www.physicspages.com/2013/03/21/tensor-product-numerical-example/
First vector: position of the earth:
Ai=(1111)
Second vector: position of Mars:
B j=(55.6×109 1 1 1 )
Metric tensor:
AiB j=(55.6×109 1 1 1
55.6×109 1 1 155.6×109 1 1 155.6×109 1 1 1
)Aim: calculate the energy required to produce the deformation for the quickbridge.
Numerical example of the use of the stress-energy tensor.
http://www.conservapedia.com/Quantitative_Introduction_to_General_Relativity
2017/02/04.Needs more spacetime curvature than the sun is currently providing. Extract energy from the supermassive black hole at the centre of the galaxy, as it rotates, as it attracts.
What is the force experienced close to a black hole? Tensile force experienced at the centre of the rod = (standard gravitational parameter \(times) length of rod \(times) mass of rod) / (4 \(times) (radius to centre of gravity)^3 ) . Schwarzschild radius of the supermassive black hole at the centre of the galaxy is 13.3 million kilometres. Supermassive black holes have a mass of hundreds of thousands to billions of solar masses. Standard gravitational parameter of supermassive black hole = gravitational constant \(times) 10 million \(times) mass of the sun = 6.67408 \(times) 10^(-11) \(times) 10^7 \(times) 1.989 \(times) 10^30 = 1.327474512 \(times) 10^27.
Force experienced at centre of rod = (1.327474512*10^27*1*1)/(4*1^3) = 3.3186863*10^26 Newtons.
How do we harness this force? Use another renewable source of energy to exert an opposite force.
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How do we harness the force around a rotating black hole? Use a gravitational turbine.
Can infinite amounts of energy be generated by a sequence of fusion and fission that repeats itself?
2017/02/07.Energy absorber absorbs gravitons. Water may absorb gravitons. Energy spewed from a material which curves spacetime the opposite way.
Use curvatures section on schwarzschild solution Wikipedia page to calculate energy required, using E=mc^2.
Schwarzschild coordinates are polar coordinates.
Quickbridge equation: Two diagonally symmetric stress-energy tensors, which are related to each other, one tensor in one set of equations, relation in another equation.
Relation of masses: 8 dimensional tensor describing the relation in prior flat spacetime; or 4 dimensional tensor with the position of one mass as the origin.
Solve equation with one mass. Superimpose the resulting stress-energy tensor on top of the stress-energy tensor of the other mass.
The original coordinates are integrated to find the new length between them.
Maybe spacetime stretching is different from being compressed but with spacetime mostly unaffected.
Quickbridge would need to compress space, but not compress the person travelling. The spacetime in the Quickbridge should compress together and units disappear while the person's spacetime remains the same. See whether equations predict spacetime units can disappear.
Graphing the schwarzschild solution would enable us to see whether the calculations are correct, and if our understanding of the use of the schwarzschild metric is correct.
The distant mass/energy for the Quickbridge can come from a fusion reactor.
Negative mass forms the tube of the Quickbridge, positive mass at the destination end.
Negative energy may create negative mass.
Schwarzschild wormhole Andrew Hamilton.
Can gravitational force be cancelled: the point opposite the mass on a circumference experiences negative force; on the side of the mass, use another mass to cancel the attraction outside the tube; use two rotating masses to create negative forces around a circumference, one in the centre and one outside the circle; use one mass outside the circle, so the middle is hollow.
Shielding required around circling mass. Energy sink to produce shielding. Energy deforms spacetime, so spacetime can be modified by energy sinks. People must wear energy sinks to pass through.
Energy required by Quickbridge: compare this to the energy in the sun.
Riemann tensor: four corners of a square, with x, y, z and time to position it; maybe this is not the exact description.
Maths to describe spacetime curvature.
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Coordinate system distorted + cancelling waves = Quickbridge spacetime deformation.
Unit square is spacetime without curvature, equation can describe curvature of multiple squares if the squares can be coalesced.
Point deformation of a square: centre of cube (0, 0, 0, 0), attracting point (x, y, z, t), magnitude of attracting force (m), attracting force is force required to move the centre of the cube by a multiple of its unit length.
Riemann tensor is a cube with 8 corners (vectors) and time (16 values in total).
Find evidence of theory in nature or environment.
LIGO publications.
Gravity does not get used up.
Aim: how to cancel out gravity. Use multiple point masses around the circumference.
Gravitational wave measurement: when a pendulum stops swinging.
Invert tensor mathematics.
Find a space for your gravitational force experiment with pots of water evaporating.
Experiment where small mass affects evaporation against gravity: greatly different amounts of water evaporating under cling film with one hole, regularly measure on scales, evaporation rate due to mass of water and temperature.
Can papers be published on personal websites? Depends on the journal policy; this is especially okay if the journal is open access.
Artificial source of gravitational waves. Study the inner ear to find out the nature of these gravitational waves. Current literature says that electromagnetic waves produce gravitational waves.
What are the features of this artificial source of gravitational waves: small mass, small amount of energy. Is there anywhere in nature where this phenomenon is found? No. Hypothesize something which may produce gravitational waves. Graviton circuit, like electric circuit. It would have a source of gravitational energy.
What forces appear to act against gravity? A balloon may emit gravitational waves which cancel out the gravitational waves emitted by the earth. This must be the case as the earth, independent of the balloon, still emits gravitational waves.
Artificial source of gravitational waves: object which increases and decreases in mass. If the balloon increased in mass, it would fall towards the earth. For the balloon to rise and increase in mass, it must continue to emit gravitational waves almost out of phase with those of the earth.
A hypothetical machine which could interfere with these gravitational waves so they are out of phase or in phase. Hypothesize a way in which it would do this. It can slow down the gravitational waves, so they change phase. Like water which holds fish up against gravity. Google: what slows down gravitational waves? Matter can focus gravitational waves, like in gravitational lensing, however, it has to be matter of high density.
Another source of gravitational waves far out in the solar system. Although if the gravitational waves towards the earth were eliminated, Mars would automatically travel outwards.
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2017/02/21.What mass would be required to curve spacetime so that the distance to Mars is changed from 54.6*10^9m to 0.1m?
(1-2*m/r)^(1/2) dr from r=0 to r=54.6*10^9 (distance to Mars).
((1-2*m)^(1/2))/(r^(1/2)) dr
= ((1-2*m)^(1/2))r^(-1/2) dr
= ((1-2*m)^(1/2))/(1/2)*r^(1/2)
= ((1-2*m)^(1/2))*2*r^(1/2)
0.1 = ((1-2*m)^(1/2))*2*(54.6*10^9)
m = 0.5kg
Check that the correct values have been used.
Detection of gravitons 2017/01/06.They travel in a straight line, and are able to penetrate objects. More massive objects emit more gravitons. The sum total of gravitons emitted can be found by measuring the total mass attracted. On small objects, look at bacteria attracted. Effect of gravitons on brownian motion. Aim: write a formula. Total mass (orbiting or on surface) attracted in one direction, multiplication, open bracket, distance to centre of object studied, minus distance to outside of object, close bracket. Tensor for 3D shape of the object. Aim: what if there were no objects in vicinity. Place a considerably smaller object in vicinity, look at its acceleration, distance to centre of object studied where this acceleration happens. Mass multiplied by acceleration. Aim: how does this build on previous theories. Gives a method of measuring graviton emission in one direction, previous theories are not direction-specific, though Einstein field equations are, experiment example applies to small objects. Total effective gravitons. Small molecules detected by radiation emitted after vibration. Force board which detects force exerted on it by small objects, do it above small object, then do it above empty space. Force board should be so sensitive that it detects a single graviton, nano spring (a molecule like a spring).
Quickbridge Direct Work. 2017/02/22.Quickbridge. 04/02/2017.
Sub-goal: How does fusion generate energy, what is the starting point and what is the end point?
Quickbridge. 04/02/2017.
Building the quickbridge at home. Refraction of light by a larger glass cup with water in it, compared to a smaller glass cup with water in it. The greater the mass of the glass cup, the stronger the light.
What are the possible accidents that may happen with the quickbridge? Improper positioning of moved objects, explosions.
There should be a portion of the quickbridge behind Mars so the gravitational waves emitted travel no further.
There is an attractional force between a balloon and other objects in space which outweighs the attraction to Earth.
15
If an object is light enough, it emits cancelling gravitational waves, so it floats up. Its own momentum overrides the force attracting it to Earth. The balloon's momentum comes from the spinning of the earth, like weather patterns.
How to use the metric tensor? Length of a parameterized curve vector: x^(r)=x^(r)((u)), where x^(r) is a vertically written vector and u is the parameter. This would be the original distance from Earth to Mars.
http://leto.net/x/2008/12/5-min-math-metric-tensor.html
Quickbridge. 2017/02/07.
Infinitesimal displacement element: dx^(r)=p^(r)((u))du. Basically p^(r) is another vector which is slightly larger or smaller than x^(r). We need the infinitesimal displacement element because we are trying to measure the length of the curve, bit by bit. The length of this displacement element is: ds = \(sqrt)(g_(mn)p^(m)p^(n))du. The length of the curve from u=u_(1) to u=u_(2) is L = \(integral)(\(top)(u_(2))\(bottom)(u_(1)) ds). Then substitute in ds into L. The metric tensor is required to define distance along a curve when working in a non-cartesian coordinate system, e.g. spherical. A two-dimensional parameterized curve vector gives a three-dimensional curve.
How to make gravitational waves cancel? The intervening machine needs to slow down the gravitational wave by 180 degrees. Use a more sensitive interferometer to detect whether the waves are in phase or not.
Hypothesize a method of artificially creating gravitons? Supercooling water. Into lower temperatures than before, e.g. 3.14 Kelvin. Supercooled water emits cancelling gravitational waves, so you can use this in the opposite direction of travel.
In a fusion reactor, hydrogen comes together to form helium. How does a fusion reactor produce energy, which bonds are involved?
Put numbers in metric tensor. x^(r)=x^(r)((u)). We just have a distance, and this is given by the vector (u).
Quickbridge. 2017/02/08.
Cancelling gravitational waves emitted = 1 / (\(exponential)(temperature of supercooled water)) .
The stress-energy tensor should be easy to populate with numbers, as only the energy of the originating source is required.
What numbers should we put into the stress-energy tensor? T_(11)=Joules per m^3, all other components are 0. If the mass-energy is contained in 1m^3, we only need to now worry about the mass and energy. 16 simultaneous equations are reduced to 1.
How do we find out the energy required to get to Mars? Write 2 simultaneous equations, one with no energy and current distance to Mars, the other with energy and shortened distance to Mars.
What numbers should go into the Ricci curvature tensor? It is symmetric. Calculate ricci curvature tensor. Definition of the ricci curvature tensor. Google: Ricci curvature tensor R11. It is calculated from the Riemann tensor. Google: Ricci tensor flat space. R11 = 0 in flat space. The Ricci tensor puts the metric tensor in context, the metric tensor describes two points and the Ricci tensor describes the way in which spacetime is curved already for these two points. Google: Riemann tensor for space
16
around the sun. Use the line element for the Schwarzschild metric to calculate the Riemann tensor. Do the rest on the computer.
Energy from fission is when the strong force in a nucleus is released by splitting of atoms. The energy from fusion comes from a slight decrease in mass when two nuclei are fused; the missing mass is released as energy. As both processes involve a decrease in mass, fusion and fission cannot be used in a cycle to produce infinite amounts of energy.
Quickbridge. 2017/02/08.
How do we artificially create gravitons? Put an atom next to positively charged plates, and electrons, so the atom is fooled by thinking it is part of more mass. Artificially mimic communication that comes with quantum entanglement. Higgs boson may interact with protons and neutrons to give them mass.
We can't artificially create gravitons, as gravity comes only with mass.
Use a black hole to attract Mars to near Earth.
Focus gravitational waves to one point and use that to attract Mars.
Line element for the Schwarzschild metric: c^(2)dT^(2) = (1-r_(s)/r)... Where c is the speed of light, T is the proper time, r_(s) is the schwarzschild radius, r is the radial coordinate.
Google: qmplus exercises class 3. Calculate metric tensor. Calculate metric tensor for schwarzschild solution.
Ricci tensor for the schwarzschild solution: R_(11)... This is not what we want, as the schwarzschild solution does not include energy.
Google: numerical calculation of the ricci tensor. Google: Calculate spacetime curvature from the Schwarzschild solution. How is mass of the quickbridge related to the curved space? Time, phi and theta are constant. Distance in curved spacetime = \(integral)(top: original distance)(bottom: 0)((dr)/\(sqrt)(1-2×mass/r)). Distance in curved spacetime is 0.1 metres. The original distance is 54.6×10^9 metres. Integral of (1-2×mass/r)^(-1/2). Integrate (1-2*m/r)^(-1/2) dr from r=0 to r=54.6*10^9. Use a graph to illustrate how m varies to give distance in curved spacetime.
2017/02/26.
Try the calculation again. What is the mass required to deform spacetime so that the distance to Mars becomes 0.1m? The radial coordinate is the circumference of a circle centered on the star divided by 2*pi in metres. Google: distance Schwarzschild metric.
r s=2GMc2
Where r_(s) is the Schwarzschild radius, G is newton’s gravitational constant, M is mass, and c is the speed of light.
d s2=(1−rsr)−1
d r2
Where ds is an invariant spacetime interval, a measure of a distance between two points in spacetime, r is the circumferential radius.
17
s=∫0
r
(1−r s
r)−1
dr
s=∫0
r
( rr−rsr)−1
dr
s=∫0
r
(r−r s
r)−1
dr
s=∫0
r rr−rs
dr
s=∫0
r
1− rrsdr
s=[r− r2
2 rs ]0r
0.1=54.6×109− (54.6×109 )2
2× c2
2GM
0.1=54.6*10^9-((54.6*10^9)^2)/2*(3.00*10^8)^2/(2*6.674*10^-11*M)
M = 1.8*10^37 kg.
This mass is greater than the mass of the earth. So the energy for the Quickbridge cannot come from fusion or fission to provide energy to distort spacetime. It is time to investigate the Higgs boson.
https://uk.mathworks.com/help/coder/examples/introducing-einstein-s-theories-of-relativity.html
2017/03/05.
Try the calculation again.
r s=2GMc2
Where r_(s) is the Schwarzschild radius, G is newton’s gravitational constant, M is mass, and c is the speed of light.
r s=2×6.674×1011M
3.00×108.
Wolframalpha: (2*6.674*10^11*M)/(3*10^8)
r s=4449M .
d s2=(1−rsr)−1
d r2
18
Where ds is an invariant spacetime interval, a measure of a distance between two points in spacetime, r is the circumferential radius.
Schwarzschild solution proper distance.
ds=(1−rsr)−12 dr
∫ ds=∫0
r dr
√1− rsr
s=∫0
r
(1−r sr
)−12 dr
s=∫0
r
( r−r sr )
−12 dr
s=∫0
r
( rr−r s )
12 dr
Wolframalpha:
integral1/sqrt(1 - s/r) dr = r sqrt(1 - s/r) + 1/2 s log(2 r (sqrt(1 - s/r) + 1) - s) + constant
Where log is the natural logarithm.
x is M.
54.6*10^9*sqrt(1 – (4449x)/(54.6*10^9)) + (½)*4449x*log(2*54.6*10^9*(sqrt(1 – (4449x)/(54.6*10^9)) + 1) – 4449x)=0.1
5.46×10^10 sqrt(1 - 8.14835×10^-8 x) + 4449/2 x log(1.092×10^11 (sqrt(1 - 8.14835×10^-8 x) + 1) - 4449 x) = 0.1
Wolframalpha ran out of a computation time.
Try Sagemath. x and y are M.
x = var('x')
solve([5.46×10^10*sqrt(1 - 8.14835×10^-8*x) + 4449/2*x*log(1.092×10^11 (sqrt(1 - 8.14835×10^-8*x) + 1) – 4449*x)==0.1], x)
Trying again in Sagemath, with the correct syntax:
y = var('y')
solve([5.46*10^10*sqrt(1 - 8.14835*10^(-8)*y) + 4449/2*y*ln(1.092*10^(11)*(sqrt(1 - 8.14835*10^(-8)*y) + 1)-4449*y)==0.1], y)
[y == -1/22245*(546000000000*sqrt(-2813/34522326606*y + 1) - 1)/log(-4449*y + 109200000000*sqrt(-2813/34522326606*y + 1) + 109200000000)]
Try graphing the result with a range of r values in Sagemath.
19
M = var('M')
plot(5.46*10^10*sqrt(1 - 8.14835*10^(-8)*M) + 4449/2*M*ln(1.092*10^(11)*(sqrt(1 - 8.14835*10^(-8)*M) + 1)-4449*M),(M,-10^6,10))
M ≈−106 kg .
It is possible for negative masses to arise from the Schwarzschild solution.
Perhaps change the interval values.
http://physics.stackexchange.com/questions/130810/what-is-the-schwarzschild-metric-with-proper-radial-distance
New calculation, where R-2M measures the proper radial distance outward from the horizon.
R−2M=√r (r−2M )+2M ln [√ r2M
−1+√ r2M ] .
M_1 = var('M_1')
plot(sqrt(5.46*10^10*(5.46*10^10-2*M_1))+2*M_1*ln(sqrt(5.46*10^10/(2*M_1)-1)+sqrt(5.46*10^10/(2*M_1))),(M_1,-10^2,10))
When mass is -10^21kg, the proper distance is 438628.000000000 - 3.14159265358979e21*I.
When mass is -10^2kg, the proper distance is 5.45999980188723*10^10 - 314.159265358979*I.
When mass is 10kg, the proper distance is 5.46*10^10.
When mass is 10^2kg, the proper distance is 5.46000019811277*10^10.
When mass is 10^3kg, the proper distance is 5.46000175086916*10^10.
M ≈−1021 kg .
Google: what is the proper radial distance with regards to the schwarzschild solution.
Calculate the proper distance for the sun.
20
R−2M=√r (r−rs )+rs ln [√ rr s
−1+√ rr s ] .
r s=3000
r=149.6×106
sqrt(149.6*10^6*(149.6*10^6-3000))+3000*ln(sqrt(149.6*10^6/(3000)-1)+sqrt(149.6*10^6/(3000)))
1.49616805081065*10^8
The proper distance is actually longer than the radial coordinate. So the smaller value (of the initial distance and distance in curved spacetime) should be r, which is the distance in curved spacetime.
Calculate the proper distance for a Schwarzschild radius of rs=10^37m and r=0.1.
s=10^37
r=0.1
sqrt(r*(r-s))+s*ln(sqrt(r/s-1)+sqrt(r/s))
1.57079632679490e37*I
s=-10^27
r=0.1
sqrt(r*(r-s))+s*ln(sqrt(r/s-1)+sqrt(r/s))
7.99277837364062*10^9 - 1.57079632679490*10^27*I
r s=−1027m.
r s=4449M .
M ≈−2.25×1023kg .
For s=10^37 and r=0.1, r=<2GM, so the proper distance is contained by the black hole. This gives an interior Schwarzschild solution. This means that the black hole itself will contain the distance from Earth to Mars.
2017/03/08.
Equation for the decay of the Higgs boson.
What does the Higgs boson do? The Higgs boson clusters around particles as they move through the Higgs field. The more mass a particle has, the more Higgs bosons it will attract.
http://edition.cnn.com/2011/12/13/world/europe/higgs-boson-q-and-a/
When does a Higgs boson decay? Almost immediately after they are produced. One possible mode of decay produces two photons.
21
http://particleadventure.org/the-higgs-boson-decays-into-other-particles.html
What creates the Higgs boson? When the Higgs field becomes excited.
https://en.wikipedia.org/wiki/Higgs_boson
Google: equation for higgs decay to two photons. (TO DO)
http://www.hep.lu.se/atlas/thesis/egede/thesis-node17.html
ΓH→γγ
What does H right arrow gamma gamma mean.
It means the Higgs boson decaying into two photons.
2017/03/24.
Write a new equation for a local field in a black hole, created by Higgs bosons, which can avoid spaghettification within the black hole.
force exerted by a single higgs boson.
To create a Higgs boson, just do what they do at CERN, smash particles together.
2017/04/14.
What area of maths do you need to know to become familiar with tensors? Differential geometry.
Metric tensor: four dimensions across, four dimensions down. Find a dimension, see how it is affected by the value in the metric tensor, out in the same dimension (as the metric tensor is symmetric).
How do you build the Quickbridge? Hypothesize the way it works, and see if that matches up to experimental data, revise the theory. Once the theory is complete, it can be built.
Cooled superconductor and magnet emit cancelling gravitational waves; increase the size of the superconductor, move it behind Mars and use it to guide the planet.
How does gravitational attraction arise? The nuclei of atoms in close proximity communicate with each other. If they are fooled into thinking they are part of a larger mass, artificial gravitational attraction occurs. Two atoms together have greater gravitational attraction than if they were summed. It stands to reason that gravity is mediated mainly by the nucleus, as this is the main provider of mass. It may be possible to send gravitational waves to a nucleus and fool it into thinking that it is part of a larger whole, thus creating artificial gravity, however, this does not fit with observations. The nuclei of atoms may communicate with a yet undescribed particle, to give rise to gravitational attraction.
Can information travel through the Einstein-Rosen bridge.
How do we solve spaghettification? A black hole may be split apart by an oasis of non-distorted space, if artificial manipulation of gravity existed.
22
Origin of the graviton: How particles communicate with each other.
Write an equation for the evaporation of water. T, S_(e), V, S, D, M_(w), M_(E). dM_(w) = (T*S_(e)*M_(w)*S*M_(E))/(D^2). Where T is temperature, S_(e) is surface area exposed to evaporation, S is surface area of water, D is distance to centre of earth, M_(w) is mass of water, M_(E) is mass of the earth.
Is it possible to artificially produce the Higgs boson? Yes it is.
How would you use the Higgs boson to give more mass to one particle? Create a stronger Higgs field.
How much energy is required to produce the Higgs boson? A small amount.
Types of calculations: types of mathematical proof.
Find an existing equation and do something to it to prove a point.
Think of how to prove something mathematically in words, and what you are trying to demonstrate.
Aim: demonstrate absorption of gravitational waves mathematically.
Gravitational waves are absorbed when meeting destructive interference. Can gravitational waves be unidirectional? Depends on the shape of the source. Do a matrix to represent 360 degrees around the Quickbridge. Think of the shape of the graph solution. Quickbridge encased inside a hollow shell, with one hole pointing to Mars.
Read abstracts from papers.
Read the gravitational waves Wikipedia article and look up unknown variables and maths operators.
Detection of gravitational waves: Higgs boson detector at different heights, giving 3D view of a field. Observe Higgs bosons as they appear naturally.
Google: how to detect the Higgs boson.
Google: Sagemath schwarzschild solution example.
Properties of the Higgs boson.
What does dx on its own mean.
How is dx^2 different from dx.
Integration of dx^2.
What does ds^2 mean.
Schwarzschild solution distance.
Proper distance Schwarzschild solution.
The Higgs field has 4 components.
Gravitational wave detector: blocks of white light.
Creating new maths: maths needs to be backwards compatible. Make existing maths practical. Google for experimental data to contribute towards theory. Dissect existing maths manipulations, e.g. multiplication.
23
Equation for a solution to the Einstein field equations that allows the distance from Earth to Mars to be shortened.
Naked singularity metric.
Local spacetime within a black hole can be changed, so it is safe to live inside one.
Apply Higgs boson equations to the Einstein field equations.
Find a part of the Higgs theory similar to the Einstein field equations.
Su(2) and spacetime curvature tensor.
Ask questions as you work things out as part of your major advance on your phone.
Which part of Higgs theory was used in this paper? Look at the equations labelled su(2).
What is su(2)?
Try to write equations: see if a tensor in the Einstein field equations can be substituted for a matrix in Higgs theory.
End of time. 2017/02/04-2017/04/06.Look at Stephen Hawking's work.
If there is a big crunch, time will end.
End of time.
How to prolong the time we have? Stop our part of the universe expanding. Turn planets and stars back into nebulae. This applies if the universe is destined to infinitely expand.
There may be a time particle, which controls how time passes for each mass. Nothing like this exists in the literature as of 2017/04/06.
Creation of a new universe. 2017/02/22.Creation of a universe. 04/02/2017.
Rewind an expanding galaxy and expand it again, like a mini-universe inside ours.
When you collapse an expanding galaxy, does everything just fall into the supermassive black hole? That's right. Perhaps there needs to be some form of artificial intervention for a black hole to release matter. Turn it into a white hole.
Can you write an equation where the black hole turns into a white hole? A white hole is the reverse of a black hole. Energy within black hole + intervention = mass from a white hole. What is a model of how a black hole works? A black hole may have a temperature, so when this is raised, it emits matter. Raise intervention variable e.g. temperature, and the energy within a black hole is converted to mass from a white hole. If the intervention variable is below a certain amount, no mass is emitted. What maths is used to calculate thresholds? Segmented regression. Threshold value for when a black hole becomes a white hole = 1 - (current temperature of black hole) / (temperature required to become a white hole). If the threshold value is 0 or negative, a white hole is formed. Can an equation be wrote where black hole forces (or spacetime curvature) become white hole forces after the change of temperature? Stop exploring this avenue because as the black hole absorbs matter, it
24
gets hotter. What is another model for how the black hole works? Search the literature. The black hole has mass, but the singularity is of infinite density. The discrepancy in this may be due to displacement of absorbed matter to another universe.
Creation of a new universe. 04/02/2017.
Is there an explanation of how a black hole may stop transferring matter into a new universe, in accordance with our observed big bang? No existing explanation. The instant a star falls into the black hole and the mass of the black hole doesn't change, is when a new universe is made. Or perhaps the black hole stores up mass in the singularity and releases it in one go to another universe. As the black hole's singularity is infinitely dense, there is plenty of material to make another universe with.
Black holes decrease entropy. 2017/04/05.This is logical, given that matter is condensed in a black hole.
Google: does entropy decrease in a black hole.
https://www.quora.com/Do-Black-Holes-decrease-entropy
Black holes can explode, giving rise to new matter.
http://www.thephysicsmill.com/2015/06/14/speculative-sunday-can-a-black-hole-explode/
Writing new maths: combine existing maths, e.g. Einstein equations and quantum theory.
Can black holes be made to explode?
By applying a large shear force.
If a cycle of entropy decrease and explosion occurs, then we have a large amount of matter to work with.
Stackpirt2017/04/05.How does a neural network work?
The network is trained using a series of inputs and outputs. As long as the neurons in the hidden layer help produce the right output, the exact path that the information takes is flexible.
http://www.explainthatstuff.com/introduction-to-neural-networks.html
How to input words into a neural network?
Turn it into binary.
http://datascience.stackexchange.com/questions/869/neural-network-parse-string-data
Use another neural network to process the words into positive and negative news.
University Maths PuzzlesAnalyse how the answers are done, thoroughly.
Folding rectangular sheet problem. 2016/08/02-2016/08/03.http://www.qbyte.org/puzzles/puzzle01.html
25
http://www.qbyte.org/puzzles/p001s.html
If a rectangular sheet is folded so that two diagonally opposite corners come together, and the crease formed is the same as the length, what is the ratio of the longer side of the sheet to the shorter side?
Rectangular sheet on its side. A diagonal crease going from the top edge to the bottom edge, like a backward slash \. Known lengths: l = crease. Crease right angled triangle shorter right angled side = w. l^2=w^2+(crease right angled triangle longer side)^2. (1). A long diagonal going from the top left hand corner to the bottom right hand corner. (Long diagonal)^2 = l^2+w^2. (2). The long diagonal and the crease bisect each other at right angles. (l/2)^2+((long diagonal)/2)^2=l^2. (3). Make long diagonal the subject of (2): long diagonal = sqrt(l^2+w^2). (4). Insert (4) into (3): (l/2)^2 + ((sqrt(l^2+w^2))/2)^2= l^2. 1^2/4 + (l^2+w^2)/4 = l^2. l^2+l^2+w^2=4l^2. 2l^2+w^2=4l^2. w^2=2l^2. 1=(2l^2)/(w^2). sqrt(1/2)=l/w.
Correction (3): (l/2)^2+((long diagonal)/2)^2=(hypotenuse of right angled triangle formed by crease and long diagonal)^2. (6) A new right angled triangle formed by dropping an altitude from the right angled triangle with the hypotenuse as the crease. This triangle is similar to the parent right angled triangle. In fact, there are two right angled triangles formed by dropping this altitude. The bigger right angled triangle from dropping the altitude. Crease right angle triangle hypotenuse: l. Crease right angled triangle longer side: w. Larger of the right angle triangle from dropping the altitude, hypotenuse: w. Larger of the right angled triangle from dropping the altitude, longer side: a. l/w=w/a. (5). Smaller of the right angled triangles from dropping the altitude, hypotenuse: unknown side of the crease right angled triangle. Smaller of the right angled triangles from dropping the altitude, longer side: b. l/w=(unknown side of the crease right angled triangle)/b.
Correction (1): l^2=w^2+(crease right angled triangle other side)^2. (9)
(9) and (6) are similar right angled triangles. Hypotenuses:longer sides = l^2/w^2 = (hypotenuse of right angled triangle formed by crease and long diagonal)^2/(l/2)^2. (7) l/w=(hypotenuse of right angled triangle formed by crease and long diagonal)/(1/2). l/w=2*hypotenuse of right angled triangle formed by crease and long diagonal. Hypotenuse of right angled triangle formed by crease and long diagonal = l/(2w). (8) Insert 8 into 6: (l/2)^2+((long diagonal)/2)^2=(l/2w)^2. (10) Insert 4 into 10: (l/2)^2+((sqrt(l^2+w^2))/2)^2=(l/2w)^2. l^2/4+(sqrt(l^2+w^2))^2/4=l^2/(4w^2). l^2/4+(l^2+w^2)/4=l^2/(4w^2). l^2+l^2+w^2=l^2/w^2. 2l^2+w^2=l^2/w^2. (11) w = sqrt((sqrt(1(1+1)))-1) = sqrt( (sqrt(2))-1) = 1.55 . (2l^2)/w^2+1=l^2/w^4. l/w=ratio. l=1, w=1.55, 1/1.55=0.65.
2l^2*w^2 + w^4=l^2. w^4 + 2l^2*w^2 – l^2=0. x=w^2. x^2+2l^2*x-l^2.
w2=−2l2±√4 l4−4 (−l2 )
2 =−l2± √4 l 4+4 l22 =−l2± √4 l2 (l2+1 )
2 =−l2±2 l√(l2+1 )
2
w2=−l2±l √( l2+1 )
Comparison with answer:
b=(a/b)*sqrt(a^2+b^2). b^2=(a^2/b^2)(a^2+b^2). b^2=a^4/b^2+a^2. b^4=a^4+a^2*b^2.
Triangular area. 2016/08/11.http://www.qbyte.org/puzzles/puzzle01.html
AEXD area = ∆ABC – (a+b+c).
26
AEXDarea=12BC∗BA∗sin B−(a+b+c)
∆ a+∆b=12BC∗BE∗sinB
AEXDarea=a+bBE
∗BA−(a+b+c)
a=12BE∗BX∗sinEBC
Divide into squares. Use calculus.
Volume of an elliptic cone with the tip removed. 2017/01/26.V=πabl .
Where V is the volume of an elliptic cylinder, a is the shortest radius of the ellipse, b is the longest radius of the ellipse, l is the length of the cylinder.
The volume of an elliptic cone with the tip removed is the volume of the smaller elliptic cylinder, plus half of the difference between the smaller elliptic cylinder and the larger elliptic cylinder.
V 2=(∫0
l2
π a2b2dz−∫0
l
πab dz)÷2+∫0
l
πabdz .
MiscellaneousFinding volume of an irregular solidSeparate into planes of a determined thickness, find the equation of the line of the irregular edge. Split the irregular line up and use parabolas. Two areas next to each other of tiny thickness: find the average of the irregular line, or the average of the area of the two areas. Curve fitting. Average of irregular line: set y as the subject and add the right hand sides together. Do programming with an irregular solid as an example. Equation of irregular line: use tangents, average tangents to an irregular line. Irregular solid's centre should be the origin, and areas calculated as if they were in the upper right quadrant. Aim: what if there were holes in the solid: need to reposition axes so they are centered in the middle of the hole.
Using radiation to move molecules in the bodySend radiation of a particular frequency to excite molecules of a specific structure, find out where the target molecules are. Application in healing. To move the molecule, make water vibrations behind the molecule, dampen vibrations in front. Or make the molecule vibrate more in one direction than it does the opposite direction. Nanotechnology to make a molecular engine.
Connect software to hardwareLook at raspberry pi projects. The software outputs to pins on the computer board. The pins are connected to the hardware.
Use text editor for new versions of the problem cruncher.
27
ErfoErfopaper. 2017/02/03-2017/02/04.First product: infinitely recyclable paper.
Aim: has anyone else done this? A search for infinitely recyclable paper yields a search result which says paper can be recycled 7 times before its fibres get too short.
Aim: what are the desirable properties of erfopaper? Lightweight (can be made into thin sheets, holds together as a thin sheet), writable (slightly rough, pale, attracts ink), nontoxic, recyclable infinitely (made of e.g. chains, which do not break down in the recycling process), cheap.
What chemicals have these properties? Nontoxic: polymer. Writable: plastic with a roughened surface. Biodegradable plastic, which degrades into carbon dioxide and water by the action of microorganisms e.g. starch-based bioplastic. Work out the entire production process so that no fossil fuels are used, so that biodegradable plastic is carbon neutral.
https://en.wikipedia.org/wiki/Biodegradable_plastic
Aim: is paper already biodegradable? No, because of the inorganic component of paper. So far, there are biodegradable paper cups, and biodegradable Gascogne kraft paper, but because paper may go to a landfill site, the paper never truly biodegrades. Stone-based papers are not truly sustainable because they require the extraction of limestone, a form of digging.
http://www.paperonweb.com/A1048.htm
http://eq.tumblr.com/post/7044176102/is-paper-biodegradable
What is required to make a sustainable biodegradable paper? Environmentally friendly processes must be used in the making and degrading process.
What advantage does an infinitely recyclable paper have over a biodegradable paper? We can see directly that paper, when recycled, becomes paper again. Whereas with biodegradable paper, we know that it becomes carbon dioxide and water, but cannot see when it eventually finds its way back into a plant. Also the biodegrading process takes time, whereas if the paper is recycled, the cycles of use can be faster.
What makes paper writable? Use nanotechnology to produce small pores on the surface of erfopaper. Let the polymer be pressed against metal with pores on it, e.g. metal membrane. The porous metal is made by dissolving hydrogen in molten steel.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3035924/
http://www.metalmembranes.com/
What is a directional pore? A pore which allows substance through it in one direction.
Is PVC 100% infinitely recyclable? Yes, via feedstock recycling.
http://www.recovinyl.com/pvc-recyclable-material-ideal-reprocessing
http://www.sustainability-ed.org.uk/pages/pvc3-3.htm
2017/02/04.Erfosurface: see if crayola will pay for advertising their brand in conjunction with this product.
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Use an infinitely recyclable plastic. Currently heating plastic causes it to degrade, meaning eventually after a certain amount of heat cycles, it becomes unusable.
Carbios.
Extract plastic from landfill waste and recycle it.
Aim: find a good way to sell enzymatic degradation of PET.
New material that requires only heating to recycle: polymer bonds don't break with heating, strength of intramolecular bonds far greater than intermolecular bonds. Look up bond strength. Organic polymer list. PolyVinylChloride feedstock recycling can result in new PVC.
Pad: each page sticks out from underneath its cover, to allow for page turning.
Make one, make more if you receive orders.
Make it at low cost so it is competitive.
Source, design, manufacture erfosurface.
Experiment with plastic pockets: smudging with another pocket, writing multiple lines.
Aim: prevent lines on erfosurface from smudging. Add double sided stickers to glue the cover to the white PVC sheet. Add pocket to the cover so the white PVC sheet is slotted in there. Sell one white PVC sheet at a time. Put PVC sheets in a hard PVC cover which opens like a book and clicks shut.
Hardness of PVC depends on the length of chain.
Make a thick pad.
Needs to be taken apart to separate PVC of different colours.
Type of printer ink which can be used to print on white PVC.
Nanotechnology to hold covers in place on white PVC.
2017/02/04.Infinitely recyclable materials.
Think of the molecular structure of existing materials.
Think of what properties you want the paper to have. Think of what would be in the molecular structure of erfopaper to give it these properties.
Process design.
Have a good business model, so erfopaper becomes popular.
Polymer which does not stick to porous steel plates.
White opaque polymer.
Fixed amount of polymer which is ladled onto porous steel, then spread out to cover the entire porous steel plate. Heat the polymer in a pot.
Scrub off pencil and soluble ink from erfopaper using a sponge and water.
Think of how the manufacturing process can be automated.
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The polymer must clearly get the pore imprint from the steel. It should be heated to high enough temperature.
Write detailed plan when choice of polymer is made.
How will the paper work in printers.
The ink must not be rubbed off easily. It should be partially absorbed by erfopaper. Use cover for paper.
Permanent copies of text can be scanned.
Erfopaper pad.
Crowdfunding or self-funding to get the pad made, then venture capitalist funding to move into manufacturing.
Making the pad: stiff plastic backing and white sheets stitched together. Binding machine.
Use washable ink from a fountain pen.
Pad cover design: PVC lock which is hinged to cover, two sides of the lock push together and lock tightly, front cover has one side of the lock, back cover has other side.
How to engineer two sides of lock? Plastic pop buttons. Triangular tongue supported by 2 strips of plastic connecting two sides of the tongue together, which means it acts like two clothes pegs and can flatten. The tongue slots into the other side of the lock.
Visit places to borrow equipment necessary to make erfosurface, for demonstration product. Work out how to automate production before first order is placed.
See if you can get by at home by "making" the product: permanent marker, plastic resealable bag, white paper in the bag.
Mum doesn't like this idea.
Erforecycler. 2017/02/04-2017/02/26.How to change one type of atom to another: nuclear fusion and nuclear fission.
Is there any need to investigate this? Nuclear fission is already cheap enough. Nuclear fusion is currently being investigated by a whole variety of people.
The energy required by fusion, and fission, for recycling can be created by a process which produces pure elements.
Sustainable goods2017/04/06.
How is farming sustainable.
The nutrient cycle.
Renewable energy from a black hole. 2017/04/08.Harness energy from a radiating black hole.
App for the visually impaired. 2017/02/06.A software which can describe a picture to a blind person in words.
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