Reflections about the maths daily in the world around us.
The nummolt - mathcats materials, the mathematics behind them and related to them.
Math education and apps.
Primes as the basic building blocks of numbers
In a mirror anamorphosis, a conical or cylindrical mirror
is placed on the distorted drawing or painting to reveal an undistorted
image. The deformed picture relies on laws regarding angles of
incidence of reflection.
The app (Conic Anamorphism Cameras) has 2 type of cameras:
1.- For to build a conical anamorphosis,
2.- for to reveal the undistorted image.
Image distortion and reconstruction can be performed using 9 different angles of cone:
From 41.4º to 75.5º :
(An undistorted triangle is a trifolium)
Some examples made with the app:
Thonet Chair:
Undistorted Thonet Chair:
Cage Bird:
Cage Bird Conical Anamorphism with center in the yellow bird:
Delaunay Triangles and Voronoi polygons in Voronoi Football / Soccer:
Delaunay triangles are the result of a triangulation such that no point in each valid triangle is inside the circumcircle of any other triangle.
In the app, each player is the vertex of a triangle
Connecting the centers of the circumcircles produces the Voronoi diagram:
The dual graph for a Voronoi diagram (in the case of a Euclidean space with point sites) corresponds to the Delaunay triangulation for the same set of points.
In this app, the Voronoi polygons are the basis of the automatic operation of the game.
The position of the players can be modified, dragging them to a new situation unless they have some task entrusted to them by the team.(mainly the player who is in possession of the ball in an attacking situation, or the goalkeeper in a defending situation)
A sample of the development of the game:
Geometry of each moment is what drives this game:
The players decide the best pass from the analysis of the edges of the Delaunay triangles. And when they advance with the ball, they usually send it to the more advanced point of their Voronoi region.
Help tool to solve problems graphically linear, quadratic, cubic and more....
TRAINS CROSSING: A train leaves Washington at 5 p.m. and
arrives in New York at 9 p.m. Fast train leaves New York at 6 p. m. and
arrives in Washington at 9 p. m. What time do they cross? At what place
of the travel?
TRAINS
CHASING: A train leaves New York at 5 p.m. and arrives in Washington at
10 p.m. Fast train leaves New York at 6 p.m. and arrives in Washington
at 9 p.m. m. What time does it reach the first one? Where on the trip?
WATER
TANK: The main faucet fills the pool in 5 hours, a second auxiliary
faucet fills it in 8 hours and the drain empties it in 10 hours. If we
leave the faucets and the drain open, in how many hours will the pool
fill up?
PAINTERS: A painter would paint the walls of a house in 8
hours. A second painter would paint them in 12 hours. How many hours
would it take the two painters to paint the house?
CLOCK HANDS
OVERLAPPING: The hands of a clock overlap many times every 12 hours. At
what point do they overlap for the first time after 12 o'clock? And the
following?
AGES: The ages of two people add 18. The multiplication of the numbers that correspond to their ages is 56. What are their ages?
GARDEN:
A small garden measures 7m. by 11m. We add around a perimeter path of
fixed width. The garden with the path has grown 63m² How wide is the new
perimeter path?
SQUARE GROWING: If the side of a square grows
4cm. and still is a square, then the area grows 64cm². Which was the
original side size of the square?
NUMBERS: A number multiplied by the next number is 56. What are the numbers?
BOX: We want to build a 3 cm high square box containing 48 cm³. How long will the side of the base be?
CUBOID:
We have a cube, and we make it grow 1m. in the first dimension, 2m. in
the second dimension and 3m. in the third dimension. The original volume
has increased by 52m³. What was the side of the original cube?
DIRECT RULE OF 3: We need 3 cans of paint to paint 2 rooms. How many cans of paint will we need to paint 6 rooms?
INVERSE
RULE OF 3: 2 big printers print and bind 1600 books in 8 hours. How
many big printers would we need to print and bind 2400 books in 6 hours?
TRAPEZOID:
The parallel faces of a trapezoid measure 3 and 9 and the distance
between parallel faces is 7. Split the surface of the trapezoid in two
of equal surface with a parallel line to both already parallel. How far
is the dividing line from the shorter parallel face?
Here is a cube made of magnesium and wolfram:
Density of Mg = 1.7
Density of Wolfram = 19.3
Both cubes weight the same.
Size of the tiny cube made of wolfram?
Doubling the cube: "When suffering from the plague of eruptive typhoid fever, consulted the oracle at Delos as to how they could stop it. Apollo replied that they must double the size of his altar which was in the form of a cube".
(Eulerian path and Hamiltonian path in a Hypercube)
Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once.
Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once.
For the existence of Eulerian trails it is necessary that zero or two vertices have an odd degree.
Determining whether Hamiltonian paths and cycles exist in graphs are NP-complete.
Euler & Hamilton App:
From the list of preloaded graphs, or those created with the edit function, the app calculates if they have an Eulerian or/and Hamiltonian path.
In each case, it only allows to start at the nodes where it is possible to make each of the paths, and marks them in green:
(Eulerian Graph with a bridge Starting nodes In green)
If there is a Hamilton path, the app calculates the shortest path. Let the user search for the shortest Hamilton path, and if not found, mark the excess distance in red.
If there are difficulties to find the shortest path, the edit mode can show the best solution:
Shortest path: ACEFBD
The editor mode allows you to create new graphs, which can be permanently incorporated into the list of graphs to study.
A geoboard is a mathematical manipulative used to explore basic concepts in plane geometry such as perimeter, area and the characteristics of triangles and other polygons.
It consists of a physical board with a certain number of nails half
driven in, around which are wrapped geo bands that are made of rubber.
Normal rubber bands can also be used.
Geoboards were invented and popularized in the 1950s by Egyptian mathematician Caleb Gattegno (1911-1988).
In my geoboard you can build all kinds of polygons and try to guess the surface they contain.
