[Goanet] {Dilip's essays} Year of the tetrahedrals and triangles

[Dilip D'Souza]

February 15 2024

Put it down to a fair amount of travel in these first few weeks of 2024,
including a fabulous birdwatching trip in the Sundarban Tiger Reserve in
West Bengal: I'm so far behind on sending out my Mint columns here that I
had lost track of the backlog.

So let me try to make amends, a few at a time.

Speaking of 2024: consider yourself lucky to be alive this year (and
reading this, why not?). Because this is a tetrahedral year, and the last
one was 1771, and the next one will be 2300. I don't think you'll be alive
for the next one.

But that sad realization apart, what is a tetrahedral number, in this case
2024? Well, they are built on triangular number, which in turn are built on
the integers. All of which hints at one of the fundamental things about
mathematics: patterns.

But there's more to discover too. Including something that applies to this
year that may apply even better to next year (remind me then).

Take a look at my Mint column for Jan 5 2024: Year of the tetrahedrals and
triangles,
https://www.livemint.com/opinion/columns/2024-year-of-the-tetrahedrals-and-triangles-11704384732215.html

cheers,
dilip

---

Year of the tetrahedrals and triangles


Meme that's making the rounds this week: This new year, 2024, is
tetrahedral.

What does that mean? No, it has nothing to do with those containers various
juices and milk and the like come in. Then again, maybe it does. Tetrapaks
were originally crafted as tetrahedrons - thus their name - but at some
point they morphed into today's ubiquitous cuboids.

Before your eyes glaze over, let me explain. Cuboids are box-shaped objects
- tetrapacks, sweet boxes, cartons, like that. Tetrahedrons are pyramids on
a triangular base. This means they have four sides - the triangle at the
bottom and, rising from its sides, three more triangles that meet at a
point on top. The earliest tetrapacks had this shape. I'm not sure why or
when they changed, but you can still occasionally buy things in a
tetrahedral tetrapack.

With that pyramid in mind, imagine a small experiment to build a
tetrahedron. Start with one small marble, that you place on a table. Not
much of a tetrahedron there, but indulge me as I call it that: looks like a
sphere, but imagine it as a tetrahedron. Now pick up that marble and lay
down three more, touching each other to form a triangle. That's the
triangular base. Place the first marble on those three, forming a two-level
pyramid using 4 marbles. There's a tetrahedron again, and this time you can
certainly discern the shape. Four identical equilateral triangles form the
four sides.

Next, a one-step bigger pyramid - three levels. Lay down six marbles to
form an equilateral triangle. On top of it, place three more as above,
forming a smaller equilateral triangle. Finally, a single marble on top.
Once more a tetrahedron. Once more, four identical equilateral triangles -
bigger ones, of course - form the four sides. You've used 10 marbles this
time.

There's a series forming here: 1 marble, then 4, then 10. 1, 4, 10 ...
What's the next number? That's the number of marbles you need for a
four-level pyramid. You will start with a base that uses 10 marbles, for a
tetrahedral total of 20.

That is, 20 is the fourth tetrahedral number. Keep going like this,
building ever taller pyramids, and eventually you'll have crafted a pyramid
that uses 253 marbles at the bottom and rises 22 marble levels from there.
That gives us the 22nd tetrahedral, which is ... yes ma'am, 2024.

Savour that. It's probably been at least 150 years since the last human
died who was alive in the last tetrahedral year, 1771; and it will probably
be at least 170 years until the first human is born who will see the next,
2300. So you and I and the rest of humanity today are rather lucky to live
through a tetrahedral year: this one, 2024.

Still, even with my suggestion to savour it, I imagine all this doesn't do
much for you. These are just more numbers, after all. Except,
mathematicians know that when it comes to numbers, there are endless
patterns to discover - which is what fuels their research efforts. So with
tetrahedral numbers.

For example, take the series of numbers I've also touched on above, that
make up the successive triangular bases of the pyramids. That series goes
like this: 1, 3, 6, 10 ... They are called the triangular numbers. Of
course, the 22nd entry there is 253, the base of the tetrahedron that has
2024 marbles.

Look at that series like this. You have crafted a given tetrahedron - say,
the two-level pyramid with four marbles. To now produce the three-level
pyramid, you need your existing one to sit on top of a triangular base made
up of six marbles. That is, the third tetrahedral number is the sum of its
triangular base and the second tetrahedral number. More generally, any
given tetrahedral is the sum of its triangular base and the previous
tetrahedral.

That will persuade you that the triangular numbers are the successive
differences between the

[Goanet] {Dilip's essays} The point where Aditya stays put, almost

[Dilip D'Souza]

February 15

India sent a craft, Aditya-L1, hurtling through space a few months ago. It
will study the sun. That's interesting by itself. But it also aims to
settle at - or actually, in a small orbit around - a particular spot known
as a Lagrangian point.

There are five such points. (Or strictly, five in relation to the Earth).
What they are, why we would send a spacecraft to one of them - these are
the kinds of questions that make astronomy so endlessly fascinating.

That's what prompted my column for January 12: The point where Aditya stays
put, almost,
https://www.livemint.com/opinion/columns/the-point-where-aditya-stays-put-almost-11704969237649.html

Please send me any reactions!

cheers,
dilip

---


The point where Aditya stays put, almost

Perhaps you've noticed: The word "Lagrangian" has been popping up in Indian
news reports over the last several days. The last time it did so, but I
suspect far less frequently, was in late 2021. That was prompted by the
launch of the James Webb Space Telescope (JWST) in December that year.
Today, the word is reappearing because India has just launched Aditya-L1, a
spacecraft meant for the study of the sun and its atmosphere. Today it is
reappearing because that "L1" means something - that Aditya will be
stationed at the first Lagrangian point. JWST, for its part, is positioned
at the second Lagrangian point, L2.

Well, not quite in either case, but I'll come back to that. So to begin
with, what are these Lagrangian points and how many are there?

First of all, they are named for the 18th Century mathematician
Joseph-Louis Lagrange. Born in Italy, he later became a naturalized
Frenchman. He made a number of important contributions to mathematics - in
calculus, solving differential equations and more. Famously, he proved his
"four-square theorem": that every natural number - the non-negative
integers - can be expressed as the sum of four squares.

But Lagrange also applied his considerable mathematical talents to
celestial mechanics. How do objects in space act on each other? What can we
say about their motion relative to other objects, influenced by other
objects?

There is the "two-body" problem, which seeks to find answers to the
questions above when it's just two massive objects we're concerned with.
How do we determine how they will move as their gravities affect each
other? If they are widely separated, can we predict the way they behave? If
they come sufficiently close to one another, will they start orbiting each
other? What if one is much heavier than the other - like our Earth and the
nearby and much smaller Moon?

The two-body problem, first solved by Isaac Newton in the 17th Century,
seeks to answer such questions about these celestial behaviours. But while
it can explain pretty accurately how the Earth and its Moon move, there are
actually more bodies involved. The Sun, for example, is a large elephant in
that particular room. What effect does it have?

So if all that sounds complicated, Lagrange applied his mind to the
possibly even-more complicated three-body problem - the Sun, Earth and Moon
forming an obvious example. Specifically, he wanted to determine if three
bodies of different masses, moving at different velocities, could orbit
each other while also staying "stationary" relative to each other. In other
words, in the same positions relative to each other. Meaning, can their
respective centrifugal forces and the gravitational pulls they exert on
each other cancel out so that they are stable in those positions?

There are ways to understand what Lagrange was thinking about, even if they
are rather different situations. For one, imagine three strong bar magnets
that you place close together on a table, each South pole pointing at the
other two. Naturally, the magnets will mutually repel: if you place them
very close, they will rapidly move backward. Place them ever further apart,
though, and eventually there is a point at which the repellent magnetic
force is cancelled out by the friction between magnets and table.

Or take the recent story about the iPhone that fell out of a Boeing
aircraft at 16,000 feet - and survived. How did it survive? Wouldn't the
phone keep accelerating and eventually hit the ground at a fearful speed
that would smash it to smithereens? Not quite. As a Washington Post news
report put it, "Any object falling toward Earth will reach a point, known
as its terminal velocity, where the force of gravity can't accelerate it
anymore because of resistance from the air in the atmosphere." That is, the
force of gravity is cancelled out by the air resistance. (Yes, that the
instrument landed in grass, and not on concrete, also helped it survive
intact.)

About three bodies in space, Lagrange was asking something analogous to
these situations: essentially, how do you balance competing forces? For the
purposes of JWST and Aditya, he solved a more restricted three-body
problem, in which two of the bodies are much larger than the third.

[Goanet] {Dilip's essays} Year of the tetrahedrals and triangles

[Dilip D'Souza]

February 15 2024

Put it down to a fair amount of travel in these first few weeks of 2024,
including a fabulous birdwatching trip in the Sundarban Tiger Reserve in
West Bengal: I'm so far behind on sending out my Mint columns here that I
had lost track of the backlog.

So let me try to make amends, a few at a time.

Speaking of 2024: consider yourself lucky to be alive this year (and
reading this, why not?). Because this is a tetrahedral year, and the last
one was 1771, and the next one will be 2300. I don't think you'll be alive
for the next one.

But that sad realization apart, what is a tetrahedral number, in this case
2024? Well, they are built on triangular number, which in turn are built on
the integers. All of which hints at one of the fundamental things about
mathematics: patterns.

But there's more to discover too. Including something that applies to this
year that may apply even better to next year (remind me then).

Take a look at my Mint column for Jan 5 2024: Year of the tetrahedrals and
triangles,
https://www.livemint.com/opinion/columns/2024-year-of-the-tetrahedrals-and-triangles-11704384732215.html

cheers,
dilip

-- 
My book with Joy Ma: "The Deoliwallahs"
Twitter: @DeathEndsFun
Death Ends Fun: http://dcubed.blogspot.com

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[Goanet] India’s top court bans anonymous election donations in blow to ruling party

[Dr. Vivek Pinto]

>"The country’s elections have become the most expensive in the world
,
even surpassing America’s notoriously pricey contests in some cases
,
at an estimated cost of up to $7.2 billion in 2019. The decision brings
opaque campaign financing, an issue bedeviling democracies worldwide, to
the forefront of India’s buzzing political chatter, as praise of the
judgment was heard from nearly all national parties other than India’s
ruling Bharatiya Janata Party."

>“'Funding in elections has been the root cause, the mother of all
corruption in the country,' said Shahabuddin Yaqoob Quraishi, a former head
of India’s Election Commission. 'The impact of the landmark judgment will
be instant, no doubt.'"

>"The biggest cash flow came from a political finance tool developed in
2018 that allowed corporations and individuals to donate anonymously
through the country’s state-owned bank."

>"The Association for Democratic Reforms found

 that nearly 85 percent of all donations in the fiscal year ending in 2023
went to the ruling BJP, as did almost 90 percent of corporate donations."

>"The judgment will not only disallow any new electoral bonds and those not
yet cashed in, but will also require the Election Commission to
retroactively disclose the identities of all donors since 2019 — likely to
create 'some ripples,' Chhokar [Jagdeep, one of the founding members of the
Association of Democratic Reforms] said."


>"The judgment stated that the nexus of money and politics allows economic
inequality to further political inequality and that the program violates
the right to information in the country."


>“'This judgment will revive the faith of the people in democracy, the rule
of law, and the Supreme Court,' Chhokar said, but the country’s 'black
money' won’t be 'disturbed at all.'”


>“'What gets declared is only a fraction of the total money that parties
collect,' he [Chhokar] said. 'This is not a panacea to all problems in
Indian elections.'”
---
By: Karishma Mehrotra
Published in: *The Washington Post*
Date: February 15, 2024


NEW DELHI — In what could be a serious blow to India’s richest and most
dominant political party, the Supreme Court struck down a type of anonymous
political donations on Thursday, just ahead of a national election this
summer.


The country’s elections have become the most expensive in the world
,
even surpassing America’s notoriously pricey contests in some cases
,
at an estimated cost of up to $7.2 billion in 2019. The decision brings
opaque campaign financing, an issue bedeviling democracies worldwide, to
the forefront of India’s buzzing political chatter, as praise of the
judgment was heard from nearly all national parties other than India’s
ruling Bharatiya Janata Party.


“Funding in elections has been the root cause, the mother of all corruption
in the country,” said Shahabuddin Yaqoob Quraishi, a former head of India’s
Election Commission. “The impact of the landmark judgment will be instant,
no doubt.”


[India’s opposition rallies against Modi’s rule in effort to ‘save
democracy’

]

The political financing for the BJP, which looks headed for a third term of
dominance in Parliament in the national elections, has ballooned in recent
years. The party earned $230 million and spent $103 million in the fiscal
year ending in 2022, according

 to the Association for Democratic Reforms, a nonproft working on electoral
reforms and one of the petitioners in the case.


The biggest cash flow came from a political finance tool developed in 2018
that allowed corporations and individuals to donate anonymously through the
country’s state-owned bank. Earlier, political parties had to reveal the
origins of donations above roughly $200. The government also did away with
a cap on corporate donations and requirements for firms to disclose their
donations on financial statements.


The government argued that these “electoral bonds” got rid of illegitimate
cash and used India’s right-to-privacy legislation to argue that the
program shielded donors’ political preferences.


The Association for Democratic Reforms found

 that nearly 85 percent o

[Goanet] Schedule for Friday 16th February 2024

[CCR TV]

CCR TV GOA
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You can also watch CCR TV live on your smartphone via the CCR TV App
Available on Google PlayStore for Android Platform.
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Schedule for Friday 16th February 2024

12:00 AM
Rosary - Sorrowful Mysteries

12:27 AM
Lent - A time of Grace - Talk by  Kenneth D'Sa

12:54 AM
Music - Bavarth - Fr Eusico Pereira

1:00 AM
Mass in Konkani

1:45 AM
Devachem Utor -Izaias  Avesor 40 -Vachpi Orlando D'Souza

1:56 AM
Pope's Intercessions

2:00 AM
Saibinnichi Ruzai - Dukhiche Mister

2:30 AM
Tumchim Vostram Nhoi Punn Tumchim Kallzam Pinzat - Talk by Orlando D`Souza

3:06 AM
Lighter Side of Lent - Fr Loiola Pereira

3:37 AM
Bhurgyanlem Angonn - Bhag 13

3:40 AM
Faith - Questions & answers - Fr. Ronnie D`Souza interviewed by Basil
D`Cunha

4:05 AM
Hymn - Bapa Mhojea - Fr Glen D'Silva sfx

4:13 AM
Golden Brigade - Apolinario and Winifred Fernandes

4:45 AM
Through Mary to Jesus - Eps 3 - Fr Malcolm Barreto S.J.

5:06 AM
Power of Forgiveness - Talk by Colin Calmiano

5:55 AM
Prayer - You are My Refuge Lord

5:58 AM
Divinity of Jesus Christ - Talk by Sheela Alvares

6:20 AM
Hymns - St Anthny's HS, Veroda, Cuncolim

6:25 AM
Couples Prayer (Konkani)

6:30 AM
Hymns - St Anthony's HS , Galgibaga

6:36 AM
Confession - Talk by Gaurish Naik

7:00 AM
Praise and Worship -  Cliton - Leaders camp 23

7:40 AM
Ximpientlim Motiam - Bhag 76 -Kofi Kop - Fr Pratap Naik sj

7:45 AM
Walking in the power of the Holy Spirit - Alfwold Silveira

8:11 AM
Music - Jezu Novem Ietolo - Fr Eusico Pereira

8:17 AM
Understand your faith - Did Biblical people live so long? - Rev Melito
D'Costa

8:21 AM
Hymn -St Rita's HS - Maina, Curtorim

8:27 AM
Devachem Utor -Izaias  Avesor 40 -Vachpi Orlando D'Souza

8:39 AM
Purgatory and Hell - Fr Henry Falcao

9:10 AM
Atmik Margdorxon - Ek Niyall 2 - Br. Malvino Alfonso OCD

9:26 AM
Our Father - Nepali

9:31 AM
The Eucharist in Daily Life - Talk by Godfrey Pereira

10:00 AM
Heads Held High - Infant Jesus High School, Colva

10:21 AM
Mon Bodlun Bapachea - Talk by Fr Victor

10:50 AM
Santam-Bhoktanchim Ladainh

11:00 AM
Health Matters  - Arthritis - Dr Zelio D'Mello

11:30 AM
Mass in English

12:15 PM
Daily Fash/ Jivitacho Prokas

12:18 PM
Aimorechen Magnnem

12:21 PM
Lenten Journey  -  Day 3 - Shirley Gonsalves

12:30 PM
Song : Sant Anton Tum Ochoriancho

12:36 PM
Repentance  - Talk by Mae Britto

1:02 PM
Role of St Joseph - Fr John Peter

1:10 PM
Pope's Intercessions

1:15 PM
Prayer for Healing from Cancer

1:22 PM
Bible Project - Way of the Exile

1:27 PM
Hymn - Mary our Sorrowful Motther

1:31 PM
Your Sins are Forgiven - Talk by Fr Michael Peters C.PP.S.

1:55 PM
Hymn -Keep up the Faith - Cassini Suiam

2:01 PM
Biblical Web Series - Eps 1 - DCBA

2:04 PM
Kuznantlim Zogddim Eps 6 - Polle  - Meena Goes and Julius Mesquita

2:27 PM
Special Prayer over the Sick - Joseph Vaz

2:29 PM
Blessed words of the Holy Ones -St Augustine of Hippo - Quote

2:32 PM
Prayer to the Holy Trinity - Prof. Nicholas D'Souza

2:35 PM
Kids Programme - What's on Your Mind ? - Episode 2

3:00 PM
Deivik Kaklutichi Magnneam

3:17 PM
Stations of the Cross - Konkani - Ulisses and Suzy

3:45 PM
Talk on Confession - Savio Rodrigues sj

4:00 PM
Rosary - Sorrowful Mysteries

4:27 PM
Gaionancho Jhelo - Bhogos maka - Monica Fernandes

4:30 PM
Senior Citizens Exercises - 17

4:56 PM
Daily Fash/ Jivitacho Prokas

5:00 PM
Magnneachem Jivit - Talk by Gaurish Naik

5:32 PM
Jezuchea Nimannea Sat Utrancher  Niyall - Br Malvino Alfonso OCD

6:00 PM
Mass in Konkani

7:00 PM
Angelus - English

7:04 PM
Catholic Perspective of Organ Donation - Fr Donato Rodrigues

7:24 PM
Hymns - St Lawrence HS - Agassaim

7:30 PM
Saibinnichi Ruzai - Dukhiche Mister

7:56 PM
Couples Prayer - English

8:00 PM
Xinpyentlim Motyam - Bhag 314 - Sony - Fr. Pratap Naik SJ

8:08 PM
Bhagevont Zuze Vazacho Ters

8:30 PM
Poem - Borvanso by Sandhya Fernandes

8:33 PM
Devachem Utor -Izaias  Avesor 41 -Vachpi Orlando D'Souza

8:45 PM
Ratchem Magnem

9:00 PM
Adoration 13 - St Anthony Church, Siolim

9:30 PM
Magnnem, Upas, Dhorm Dan - Anthony D'Souza

10:09 PM
Prayer for Laity - Giorgio Mazzola

10:27 PM
Dr Jorson Fernandes interviwed by Jovito Lopes on effects of coal

11:00 PM
Agree or Disagree - Road Safety  hosted by Ashley do Rosario

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