Mathematical Invention and Discoveries

1. Mathematical
Invention and
Discoveries
Prepared by:
Resuena, Marisol A.
Tejada, Junila A.

2. The Flyspeck project is complete:
We know how to stack balls!
Thomas Hales has announced a
formal proof of the Kepler
conjecture
These packings are both around
74% efficient (𝜋/18−−√π/18 to be
precise), and consist of layers of
spheres arranged in a hexagonal
grid
This technology cuts the mathematical
referees out of the verification process,” says
Hales. “Their opinion about the correctness of
the proof no longer matters.”

5. The 1998 proof is by exhaustion (checking all
possible cases), and referees have said they are
99% certain of it. The Flyspeck proof is
constructed using a combination of the formal
proof assistants Isabelle and HOL Light and
starts from basic axioms to make sure the
statement is thoroughly proved.
The announcement, which was made on
10th August, includes a description of the
methods used, as well as links to download
the entire code of the project.

7. The rest of the formalization has been carried out in
HOL Light, producing a formal theorem
|- import_tame_classification
/ the_nonlinear_inequalities
==>the_kepler_conjecture
where the_kepler_conjecture is defined as the following
term
`(!V. packing V ==> (?c. !r. &1 <= r ==>&(CARD(V
INTER ball(vec 0,r))) <= pi * r pow 3 / sqrt(&18) + c * r
pow 2))`
In standard mathematical language, this states that for every packing V
(which is identified with the set of centers of balls of radius 1), there
exists a constant c controlling the error term, such that for every radius r
that is at least 1, the number of ball centers inside a spherical container
of radius r is at most pi * r^3 / sqrt(18), plus an error term of smaller
order. As r tends to infinity, this gives the density bound pi / sqrt(18) =
0.74+, which is the density of the face-centered-cubic packing.

8. Inventor: Alexander Fleming Year: 1928
What Happened: Halfway through an experiment
with bacteria, Alexander Fleming up and went on
vacation. Slob that he was, he left a dirty petri dish
in the lab sink.
Big Discovery: When he got back, he found bacteria
had grown all over the plate, except in an area
where mold had formed.
Unexpected
Discovery:
The Penicilin

9. Mathematics is Explored in New
London Gallery
The Winton Gallery highlights the
importance of mathematics through 120
objects, including a cash dispenser, the
Enigma encryption machine, a model of a
super tanker, a revolutionary aircraft and a
19th-century instrument for measuring
tides.
Located in a regenerated old wing of the
famous museum, the gallery was co-
designed by celebrated Iraqi-British
architect Zara Hadid, who died of a
heart attack in Miami in March before
the gallery's completion.

10. The gallery incorporates
practical, occasionally comical,
applications of maths,
including a machine that
insurers used to calculate life
expectancy before the age of
computers, and another that
helped bookmakers rapidly
recalculate the odds on dog
races.
"We're telling stories about
how mathematics is at the
heart of everything we care
about, from life and death, war
and peace, money and trade,"
said Rooney.

11. "Maths is really hard,
challenging, but also
really rewarding," he
added. "We don't want
to leave people with a
sugarcoated vision of
maths, there are some
really difficult stories
here.” Zara Hadid

12. • New research by mathematics
Professor Richard Bertram has
successfully reactivated
oscillations in insulin-producing
pancreatic beta cells -- one of the
first necessary steps to
resurrecting the dormant cells and
restoring the production of insulin.
That's the problem for people living with diabetes: Their pancreatic cells either
don't make any insulin, or they don't make enough to control blood sugar,
causing dangerous hyperglycemia -- high blood glucose.

13. • The device, about the
size of a credit card, is
etched with
microscopic channels
that can deliver
perfectly measured
and controlled
amounts of a glucose
liquid solution to
dormant pancreatic
beta cells, which form
clusters called islets.

14. • With Roper's
microfluidic device,
researchers delivered
tiny doses of glucose
-- about one microliter
or 1/1000th of a
raindrop -- to the
dormant pancreatic
beta cells of mice.
When those
controlled doses were
administered in
rhythmic pulses of
exactly the right size
and frequency,
mimicking a healthy
body, they triggered
inherent oscillators in
the islet cells.

15. “We can think of this as making
progress toward that Type 2 diabetes
is a very complicated disease. The way
that we can beat it is by understanding
how all these components work and
that's what we're contributing to.
Scientific breakthroughs are often
facilitated by mathematics.”
- Richard Bertram

16. AirSelfie capture photos and videos in midair.
The AirSelfie is the brainchild of
Italian entrepreneur Edoardo
Stroppiana, who came up with the
idea in 2014.
It is equipped with a 5-megapixel
camera that can shoot full high-
definition (HD) 1080p video, as well
as a 4GB microSD card.
The drone's four rotors help it fly up to
65 feet (20 meters) in the air. The
flying camera measures only about
3.72 by 2.65 by 0.42.

17. •The AirSelfie is
controlled via a free iOS
or Android app.
•Users can also activate a
10-second timer,
AirSelfie uses Wi-Fi to send photos
and videos wirelessly to
smartphones. The app also allows
users to post photos and videos taken
immediately on social media.
The first pre-ordered drones are
scheduled for delivery in March. The
drone is expected to hit the market in
2017 for a retail price of $300.

18. Holography
• Holography
mathematics proposed
by Dennis Gabor
• • Emmett Leith & Juris
Upatnieks produced
first laser hologram
• Hologram: an optical
device using
photographic
techniques and laser
light to create a 3
dimensional image

19. Basic Holographic Setup
• Must have extremely stable, air bearing table: changes < λ
• Laser must operate in TEM00 mode
• He-Ne, Argon and Krypton most common
• Beam split into two by splitter
• Reference beam reflected by mirror through converging lens
to illuminate Photographic Plate
• May be photographic emulsion, photoresist, thermoplastic
or dicromated gelatin
• Object beam also spread by converging lens to uniformly
illuminate object so light reflects on plate
• Pinholes on lens are spatial filters to remove optical noise
from dust and defects in system

21. •
• http://medicalxpress.com/news/2016-11-
crunching-math-diabetes.html
• https://www.physics.ohio-
state.edu/~kagan/AS1138/Lectures/holograph
y-e894l15l_11-30.pdf