Monday, August 21, 2017

Plotting the circle equation in the 4th dimension (initial draft)

A quick summary of the calculations required to plot x^2 + y^2 + z^2 + w^2 = 1 on a graph. First, calculate x, y, z and w using a spreadsheet, for random values that satisfy the above equation:
Plot on a graph and verify:
The figures 2 and 3 in the centre may actually explain scattering and orbital jumping. Geometrically, an infinitesimally small point P can fall only near 1 or -1 since x^2 = 1, and therefore w^2 = 1 could only fall near 1 or -1 in 4D space.

Friday, July 14, 2017

A clear definition of n-dimensional spaces

- Ramanraj K

14thJuly, 2017


2-D and 3-D co-ordinates can be plotted with absolute certainty. However, 4-D and higher dimensions in n-D spaces are not clearly defined and their co-ordinates cannot be plotted with mathematical certainty. A definition of dimensions higher than three is necessary for clarity and use in mathematics, physics and computing. This would help both man and machine to describe and visualise virtual models of the world.

1-D Space

The one dimensional space is defined as a stright line along x-axis with a single vertex O in the centre.

Example: A railway track with no branches.

2-D Space

Two dimensional space is defined by two intersecting perpendicular straight lines x and y with a single vertex O.

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Example: The screen frame, with top and left as co-ordinates.

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3-D Space

Three dimensional space is defined by two intersecting perpendicular straight lines x and y with a single vertex O, and a third straight line z perpendicular to both x and y passing through O.

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Example: A train engine, a ball, earth, sun, and moon in space.

4-D Space

Four dimensional space is the 3-D space defined by lines x, y and z as above whose vertex O1 lies on a straight line w with vertex O. Therefore, in 4-D space at least two vertices co-exist and the distance between the vertices O and O1 is defined by the 4th dimension w. The lines x and w coincide. Let a 3-D space with vertex O1 be defined as follows:

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Then, the 4-D space is as follows:

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In the above figure, w = 4, and the distance between the vertices O and O1 is 4 units. This is useful to define the relationship between a 3-D object with a given vertex O and another vertex O1. Any number of 3-D objects may lie on line w and the distance between O and the vertices O1 to On may be defined by w.

Example: A train engine defined by 3-D co-ordinates on a track, with no branches. The track is the 4th dimension, and if a station is located on the vertex of the straight line, then the distance between the train and the station is given by the 4th dimension.

5-D Space

Five dimensional space is the 3-D space defined above whose vertex lies in a 2-D plane. Therefore, in 5-D space at least two vertices exist and the distance between them is defined by the 4th and 5th dimensions.

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6-D Space

Six dimensional space is the 3-D space whose vertex lies in another 3-D space. In 6-D space at least two vertices exist and the distance between the two is defined by the 4th, 5th and 6th dimensions.

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7-D Space

In seven dimensional space, at least three vertices exist, with a six dimensional space having at least two vertices O1 and O2 where O1 lies along on a straight line s with vertex O. Let the following be a six dimensional space:

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Then, the 7-D space is as follows:

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8-D Space

In eight dimensional space, at least three vertices exist, with a six dimensional space having at least two vertices O1 and O2 where O1 lies along on a plane with vertex O, and axis s and r.

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9-D Space

In nine dimensional space, at least three vertices exist, with a six dimensional space having at least two vertices O1 and O2 where O1 lies on the 3-D space with vertex O and axis s, r and q.

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n-D Space

The above pattern can be nested to infinite levels, in sets of three. The sun, earth and moon can be plotted without any ambiguity in nine dimensions. Other dimensions may be added in sets of three, down to fundamental particles, or higher up to other galaxies, and their relationships may be mapped and studied with more clarity.

For example a 12-D space would be as follows:

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A higher number of dimensions can always be expressed in lesser number of dimensions. For example, no matter how many levels deep, they can be plotted in a 2-D plane of the screen, or a 1-D line of bits and components.

Arrays can be used to efficiently store information about the dimensional space. The dimensions higher than 3 spatially relate to mod 3 and can be represented as a graph with parent and child nodes. Any two nodes may be singled out as frames of reference.

Last modified: Fri Jul 14 05:21:45 IST 2017

Sunday, December 15, 2013

Strengthening Democracy

The results of the assembly polls held in December, 2013, in four states are as follows:

Total Seats/
Seats Obtained
Majority Seats Required/
Vote Share Obtained
Madhya Pradesh230116

Two observations:

[1] No party has obtained 51% of the vote share in any state.
[2] In Delhi, no party can form an effective government, given that it is difficult for BJP, AAP, and Cong to form a coalition government [going by statements made as this was written].

It is ideal and desirable that the ruling party has 51% of the vote share in a state and a hung assembly is made impossible.

Two amendments in the law are required to ensure this:

[1] The number of total seats in a state should be an odd number.
[2] If any single party does not have 51% of the vote share and 51% of the seats, then retaining the seats obtained by the top two ranking parties, the rest should be declared vacant, and a re-election should be conducted in respect of those seats, closing nomination for all except the top two parties.

Sunday, October 27, 2013

Venus, Australia, Moon and Earth

An article titled "Did Venus Give Earth the Moon? Wild New Theory on Lunar History" has been published recently at  Earth's moon is suggested to be a present from Venus.

My thoughts on the theory.

By some ancient accounts, the Moon was positioned between Rohini and Revathy constellations.  It was always 'full moon', bright and shining, probably a planet whose orbit was between Earth and Venus.

About 40,000 years ago, massive volcanoes must have erupt on Venus in the Ishtar Terra region, blowing a mass the size of Australia into space. Let us call this mass as 'Australia' - it impacts Moon driving it closer to Earth, as a result of which the Moon is captured by Earth's gravity.

'Australia' also orbits around Earth for some time and then crashes into Earth, east of Japan around the equator. 'Australia' settles down as Australia covering Earth with dust and debris for at least 1000 years.

Entry impact from north ...
skids and settles down

This catastrophic chain of events is very well recorded in Indian puranas and texts.

The Shiva Purana portion dealing with Daksha Yagam narrates how the Moon was wedded to Revathy and Rohini, and then later 'cursed' to spend equal time with other wives, the 28 constellations, and also to wax and wane as a punishment. For a brief spell, Moon turned completely dark and later settled down in its present orbit around the Sun, along with Earth.

'Australia', orbiting Earth, is probably described as 'Tripura' – the third.

It may also be the 'Kurma' - the boar shaped one that emerged anew on the face of Earth after the great deluge.

The debris from Venus impacted several parts of Earth. The bits which fell in India are sanctified and preserved intact as 'Shakthi Peetams' till date.

Venus is recorded to have a 'beard' or 'horn' that is not observable today. The Venusian horn must have made its way to Earth – as a gift to admirers here, knocking Moon into its present orbit on its way!

[1] has images of Ishtar Terra and details of volcanic activity on Venus. Ishtar Terra appears quite flat and resembles Australia.

[2] “Why are Things the way they are”, G. Venkataraman, 'Height of Mountains' pp. 62-67

[5] The Chronicle of Mangadu Temples, Dr. P.S. Somasundaram


Sunday, May 26, 2013

Bay of Bengal - A pristine water resource

Ian Morris, in “Why the West Rules – For Now” predicts that South India is likely to face serious water shortages by 2025. It is not too wrong to say the shortage has already arrived in 2013 itself – starting with Tamil Nadu. Three suggestions to improve availability of potable water in India.

View of Bay of Bengal from Chennai

[1] Make Bay of Bengal a fresh water storage body:

The Bay of Bengal is the largest bay in the world. This bay occupies an area of 2,172,000 KM². A number of large rivers – the Ganges, Brahmaputra, Godavari, Mahanadi, Krishna and Kaveri flow into the Bay of Bengal. It is estimated that the annual run from the rivers into the bay is about 8,000 KM3. The average depth of the bay is 2.5 KM. Then, the total volume of water is around 2172000 x 2.5 = 5,430,000 KM3. If there is no salt water incursion into the bay, it would be the largest fresh water bay in the world as well.

A few studies have revealed that a saline 'pump' near Sri Lanka adds saltiness to the water by circulating waters from the Arabian Sea into the bay. If the 'pump' is managed, then in due course, the bay will turn into a natural fresh water resource.

As such, the salinity peaks 50m below the surface. Even if 10m of the fresh water floating on the bay surface is utilizable, 217200 KM3 of fresh water would be available. It is the fresh water floating on the bay that promotes the monsoons. Salinity in the bay should be carefully measured and monitored to convert the bay into a pristine water resource.

[2] Pipe river water from North to South:

Since there is very sizable runoff into the bay, it would do no harm to distribute water from the Ganges, Brahmaputra, Godavari, Mahanadi and Krishna to the South. I doubt it is practical to “network” and “link” the rivers in any fashion. The only practical approach may be to carry water in closed pipe lines like the Veeranam line supplying 180 million litres of water daily to Chennai. The pumping lines could be laid parallel to railway tracks for easy construction, networking and maintenance.

Chennai Metrowater sells per load of 10M3 of water at Rs. 670/- A load of divine Ganges water drawn straight from Haridwar might fetch premium prices in the south.

[3] Prevent Pollution:

This is easier said than done.  Pollutants are often let into water bodies, as there is no other drain into which it could be let into. An extensive drainage system should be built so that the pollutants can be streamed into it, processed and recycled. The drainage stream ought not to be allowed to run into rivers, canals, bay or the oceans.

The shortest river that empties into the bay, is River Cooum – about 64 KM long with its estuary near the Chennai harbour.  Until 1960s, this was a fresh water river. This river is now virtually used as a drain. No river should be used to drain wastes. The real test to check if pollution control measures are working correctly is to find potable water in the Cooum.


[1] 'Why the West Rules – For Now', Ian Morris
[7] In re Networking of Rivers,

Friday, June 01, 2012

Celebrating 125 years of Nikola Tesla's Electro Magnetic Motor and Alternating Current System

On May 1, 1888, Nikola Tesla Patented his following inventions:
In the SPECIFICATION forming part of Letters Patent No. 382,280, dated May 1, 1888, on the file of the United States Patent Office, Nikola Tesla claimed as follows:
“What I claim is- The method herein described of electrically transmitting power, which consists in producing a continuously progressive shifting of the polarities of either or both elements (the armature or field magnet or magnets) of a motor by developing alternating currents in independent circuits, including magnetizing coils of either or both elements, as herein set forth.”
The birth of the commutator free electromagnetic induction motor and the alternating current system born conjoint with it, from the fertile mind of Tesla, 124 years ago, deserves grand commemoration on its Quasquicentennial falling on May 1, 2013. Though Tesla's inventions have enriched human life on a grand scale, we have not given him the honour and remembrance he deserves. This should be amended for the sake of Nikola Tesla who spent every hour of his magnificent life improving living conditions of mankind by his new and improved inventions shaped by sharp observation and deep analysis honed by his desire to offer only clean solutions all the time.
Through his patents, Tesla talks to us directly even today. The specifications for his inventions make delightful reading and the figures that accompany are works of art. We can easily imagine what he envisioned and the collection of Tesla patents on the file of the United States Patent Office is probably the most outstanding gift to mankind from a single individual in history. The inventions highlight the importance of making practical use of knowledge gained and the flow of the patents stress the need to refine and improve to the point of elegance beyond which no man can improve.
The innovations sparked by Tesla are numerous and the general direction went thus:
No. 390,413 SYSTEM OF ELECTRICAL DISTRIBUTION Patented Oct. 2, 1888
No. 396,121 THERMO MAGNETIC MOTOR Patented Jan. 15, 1889
No. 405,858 ELECTRO MAGNETIC MOTOR Patented June 25, 1889
No. 416,191 ELECTRO MAGNETIC MOTOR Patented Dec. 3, 1889
No. 416,194 ELECTRIC MOTOR Patented Dec. 3, 1889
No. 416,195 ELECTRO MAGNETIC MOTOR Patented Dec. 3, 1889
In Patent No. 416,195, Tesla introduced poly phase current system thus:
“What I claim is - 1. In an alternating-current motor, the combination, with an armature wound with closed coils, of main and supplemental field magnets or poles, one set of which is adapted to exhibit their maximum magnetic effect simultaneously with that set up in the armature by the action of the other, as set forth. 2. In an electro-magnetic motor, the combination, with an armature, of a plurality of field or energizing coils included, respectively, in main circuits adapted to produce a given difference of phase and supplemental or secondary circuits adapted to produce an intermediate difference of phase, as set forth.”
More refinements, measuring tools and gadgets followed:
No. 455,068 ELECTRICAL METER Patented June 30, 1891
No. 511,559 ELECTRICAL TRANSMISSION OF POWER Patented Dec. 26, 1893
No. 512,340 COIL FOR ELECTRO MAGNETS Patented Jan. 9, 1894
No. 514,167 ELECTRICAL CONDUCTOR Patented Feb 6, 1894
The last one, wireless transmission of energy, is yet to be realized on a grand scale, but would be the ultimate goal for mankind.
The patents have a straightforward pattern: a problem is stated and the solution found is described including scientific explanations of why and how the invention works. Tesla had clean habits and this cleanliness pervaded his work. Clean engineering is the hallmark of Tesla evident from his electro-magnetic motor designs and the alternating current systems. Every little defect is attempted to be corrected in subsequent inventions to bring about perfection. His daring spirit to build the hydroelectric power generator across the Niagara when the whole world doubted its feasibility is an example of the kind of attitude required to achieve arduous goals. The world truly misses this great scientist and engineer whose love for humanity and progress knew no bounds.
Every school and institution should make an effort to teach the life and work of Tesla to inspire and create the progressive, prosperous and peaceful world he envisioned. In India, Tesla is not taught the way it ought to be done. When I was at school, we had SUPW (short for Socially Useful Productive Work) class from 5th to 10th standard. For boys, it was Electrical Gadgets [EG], and for girls -  it was Tailoring. Of course, either could opt for the other class. Mr. Subramani, who ran an electrical gadgets store, was the part time teacher for EG. Many parents were agitated about EG and the teacher used to be extra cautious in explaining the dangers of electricity while teaching its properties and uses. The notes of every class would fit exactly one page, with just five or six points which could be remembered for life. He started the first class with the lesson on bar magnets - the points went: magnets have two poles, north and south; like poles attract, unlike poles repel, they have imaginary lines of force, and the next lesson would elaborate on the lines of force, and after a few classes there would be a project. He taught EG with so much enthusiasm and passion, making sure he did justice to his job. I would rank him one of the best teachers at school. He introduced magnets, batteries, DC, AC, copper coils, 28G wire, batteries, transformers, and a host of other electrical concepts without which I would hardly be able to appreciate Tesla. I do not remember my teacher having elaborated on 3-phase/poly-phase AC or the induction motor design which could have been easily assimilated into the EG classes. For all I know, SUPW no longer seems to be part of the CBSE curriculum and with it EG has gone too. SUPW classes should be brought back to schools. Working models of the Tesla patents should be demonstrated to students at the appropriate levels. Every school leaving child should have a working knowledge of electricity.
High level research is possible into every statement Tesla made. For example, he saw connections between earthquakes and charges passing through the earth. Every little observation is magnified and attempted to be converted into a force that could be controlled purposefully. Resonance with nature carried Tesla far into the future. Tesla patented his gainful research in his youth but as he aged, he appears to have lost faith in the system, and many findings might have died with him without disclosure. Compulsory minimum guaranteed royalties for the life time of the patent, say fixed at minimum 1.5% of the product cost, should promote the cause of inventions.
India also has special reasons to celebrate the works of Tesla, who was deeply inspired by Swami Vivekananda and Jagadis Chandra Bose. Tesla used “Akash” and other vedic sanskrit words to explain his visions. Tesla was a bridge between the earth, working with his hands and feet firmly grounded to building practical tools that worked well, and the cosmos, envisioning grand goals with his mind piercing through its every kosha - layers. The present day world needs to pay an appropriate tribute to this genius and the best that can be bestowed is by working just as hard and wise as Tesla.
One of Tesla's unpublished addresses which he himself quoted in a letter to the editor of New York Times, April 19, 1908, was as follows:
According to an adopted theory, every ponderable atom is differentiated from a tenuous fluid, filling all space merely by spinning motion, as a whirl of water in a calm lake. By being set in movement this fluid, the ether, becomes gross matter. Its movement arrested, the primary substance reverts to its normal state. It appears, then, possible for man through harnessed energy of the medium and suitable agencies for starting and stopping ether whirls to cause matter to form and disappear. At his command, almost without effort on his part, old worlds would vanish and new ones would spring into being. He could alter the size of this planet, control its seasons, adjust its distance from the sun, guide it on its eternal journey along any path he might choose, through the depths of the universe. He could make planets collide and produce his suns and stars, his heat and light; he could originate life in all its infinite forms. To cause at will the birth and death of matter would be man's grandest deed, which would give him the mastery of physical creation, make him fulfill his ultimate destiny.
To ignore Tesla is to invite peril. Due appreciation will help mankind flourish.
Tesla resources:

Thursday, February 10, 2011

Pie Maths Galaxy ...

I came across "Galaxy of Mathematicians", Volume-1, published by the Pie Mathematics Association, Chennai. The book gives the brief life histories of Euclid, Archimedes, Pythagoras, Rene Descartes, Isaac Newton, Leibniz, Fermat, Gauss, Euler and Erdos along some of their important works. The life of Leibniz speaks of the 'Calculus Wars', his study of binary numbers, arrays/matrices leading to 'stepped reckoner (calculator)',and why he deserves the title, "Father of Applied Sciences". This is a good introduction to the original works of the authors mentioned.

The back cover text on the book says the publisher association is for removing mathphobia. Surely we need more such groups to grow and nurture mathematics!