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Flextangles are paper models with hidden faces. They were originally created by the mathematician "Arthur Stone" in 1939 and became famous when Martin Gardner published them in December 1956 issue of The Scientific American. Although you can find many different examples and ready to use templates on the web, the best method is to create your own template by using an interactive geometry software like GeoGebra. As a class activity creating flextangles by using a software can lead to discussions about translation and reflection. Flextangles, gizli yüzleri ortaya çıkarmak için esnetilebilen kağıt modellerdir. İlk olarak 1939'da Matematikçi Arthur Stone tarafından yaratılan flextangles, Martin Gardner'ın 1956 Aralık ayında The Scientific American'da yayınladığı makalede yeralınca, ünlü hale geldi. Webde bir çok örneğini ve taslak çizimlerini bulabileceğiniz flextangles için, GeoGebra gibi herhangi gibi geometri programı kullanarak kendi tasarımlarınızı da yaratabilirsiniz. Flextangle ları bir sınıf aktivitesi olarak program yardımıyla tasarladığınızda öteleme ve yansıma konularında da pratik sağlıyor. Ready to use Templates / Kullanıma Hazır Taslaklar: ------ ------ ------
- Net of a Sphere, Different Map Projections, anda library in Milan, Italy
Veneranda Biblioteca Ambrosiana Milano, Italy I have discovered a library while I was making my research about the lesson on spheres. We know that it is not possible to draw the net of a sphere like cylinders, cones or polyhedra. That's why it is not easy to map our our spherical world on a 2D paper. There are many different projections to map the world. You can try the interactive of Mathigon to see a few of these projections and how they distort the real size and places of the continents. https://mathigon.org/course/circles/spheres-cones-cylinders Many Mathematicians tried to converge the sphere as different polyhedra so that by using their nets, they could draw the maps. For instance, Buckminster Fuller designed his map by using triangles since he uses an icosahedron ( A Platonic Solid with 20 triangular faces) as the main shape of our world. This projection style is called Dymaxion (Fuller) projection - For the image and related article: https://en.wikipedia.org/wiki/Dymaxion_map One of the most famous polymath of the human history, Leonardo Da Vinci, used eight congruent Reuleaux Triangles* as the net of the sphere. Octant projection (1514), Leonardo da Vinci - For the image and related article: https://en.wikipedia.org/wiki/Leonardo%27s_world_map *A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself Codex Atlanticus is the name of the Da Vinci's notebook that includes this and many other drawings. To see the collection of all his notebooks, please visit https://www.discoveringdavinci.com/codexes According to the newsi Bill Gates purchased one of these books "Codex Leicester" for 30 million dollars. The notebook that we are looking for is “Codex Atlanticus” and it is original pages are in this little library in Milano Ambrosiana Library. The official website of the library and the art gallery: https://www.ambrosiana.it/en/ You may visit the Ambrosiana Library virtually with the help of Google Arts and Culture. The better news is that we can find this 1119 page - notebook online and categorized as algebra, geometry, physics, natural sciences and etc ... The online platform where you can find Codex Atlanticus is; http://codex-atlanticus.it/#/Overview Another reason that this library is a sacred place for the mathematicians is it also has the original copy of “Divina proportione” by Luca Pacioli. Leonardo's drawings are probably the first illustrations of skeletonic solids which allowed an easy distinction between front and back. For the Platonic solids, Da Vinci supplied two views: a plane view and a “vacua” or empty view where he removed the sides to better reveal the complete structure of the polyhedron. These “nets” of vertices and edges illustrate the artist’s graphic genius. Skeletonic solids Image: https://sciencemeetsfaith.wordpress.com/2019/12/14/luca-pacioli-golden-ratios/ Divina proportione
- Kürenin Açınımı, Farklı Haritalama Teknikleri ve Milano'da bir kütüphane;
Veneranda Biblioteca Ambrosiana Milano, Italy Bu kütüphane ve ona ait mini sanat galerisini keşfetmem, kürenin açınımı ile ilgili yazdığım ders sayesinde oldu. Diğer üç boyutlu geometrik cisimlerin aksine, kürenin açınımını iki boyutlu kağıt üzerinde göstermek mümkün değil. Bu sebeple, küreye yakınsayan şekli ile Dünyamızı gösteren haritalar ya kıtaların yerlerini ya da büyüklüklerini gösterirken yanlışlık yapmaya mahkum. Bunun sonucu olarak değişik haritalama metotları doğmuş. Bu konuyla ilgili Mathigon’un interaktif görselini mutlaka deneyin. https://mathigon.org/course/circles/spheres-cones-cylinders Küreyi farklı 3D cisimlere benzetip, onların açınımlarını kullanarak yansıtmaya çalışan birçok matematikçi olmuş. Örneğin geodesic dome’ların mucidi Buckminster Fuller Dünyayı düzgün 20 yüzlüye (icosahedron) a benzeterek, açınımını yani haritasını, üçgenler kullanarak çizmiş. Dymaxion (Fuller) projection - görsel ve açıklamalar için: https://en.wikipedia.org/wiki/Dymaxion_map Insanlık tarihinin en önemli bilginlerinden biri olan Da Vinci de küreyi, Reuleaux üçgenleri * kullanarak sekiz parçaya bölmüş ve bu şekilde haritayı oluşturmuş. Octant projection (1514), Leonardo da Vinci Görsel için: https://en.wikipedia.org/wiki/Leonardo%27s_world_map *Reuleaux üçgenleri, aynı yarıçaplı üç çemberin kesişmesi ile oluşan üçgenlere denir. Çember gibi, paralel iki düzlem arasında hareket ettiklerinde yükseklikleri değişmez ve sabit kalır. Da Vinci’nin bu ünlü çiziminin de yer aldığı defterinin adı Codex Atlanticus. Bu defteri de diğer bir çok çalışmasında olduğu gibi digital hale getirilmiş durumda. Da Vinci’nin tüm defterleri için mutlaka https://www.discoveringdavinci.com/codexes sitesini ziyaret edin. Bir habere göre, bu defterlerden bir tanesi Codex Leicester 30 milyon dolara Bill Gates tarafından satın alınarak kendi kişisel koleksiyonuna eklenmiş. Bizim araştırdığımız defter ise “Codex Atlanticus” orijinal hali ile Milano daki küçük ama çok ünlü bir kütüphane olan Ambrosiana Kütüphanesinde duruyor. Kütüphanenin resmi sitesi https://www.ambrosiana.it/en/ Bir bölümünü sanal olarak Google Arts and Culture daki sayfasından gezebilirsiniz. Daha güzel bir haber de bu 1119 sayfadan oluşan defterin tamamıyla dijital ortamda bulunabilmesi üstelik ücretsiz. Defterin sanal versiyonuna ulaşmak için; http://codex-atlanticus.it/#/Overview sitesini ziyaret edebilirsiniz. Bu defter 1478 ve 1519 yılları arası, Da Vinci nin geometri, cebir, fizik, fen bilimleri, icatları ile ilgili açıklamaları ve çizimleri içeriyor. Site branşlara göre arama yapmanıza olanak sağlayacak şekilde Da Vinci’nin çalışmalarını düzenlemiş. Bu kütüphanenin matematikçiler için kutsal bir yer olmasının bir diğer sebebi de ünlü İtalyan Matematikçi Luca Pacioli’nin kendisinden de ünlü kitabı “Divina proportione” ın orjinalinin burada sergileniyor olması. Bu kitap, Da Vinci’nin Pacioli’den matematik dersleri alırken, Pacioli’nin o sıralarda yazdığı İlahi Oran kitabı için, çok yüzlü cisimlere ait çizimlerini de içeriyor. Da Vinci özellikle Platonik Cisimleri resimlerken iki farklı metot kullanmış. Özellikle cismin yüzlerini boş bırakarak her yönden görünümü sağlayan çizim tekniği, bu alandaki ilk örnektir. Skeletonic solids Görsel: https://sciencemeetsfaith.wordpress.com/2019/12/14/luca-pacioli-golden-ratios/ Divina proportione - İlahi Oran Kitabı Bu kitabın siyah - beyaz ve ya renkli kopyaları da günümüzde satın alınabiliyor . Tüm bu kaynaklara Milano ya gitmeden de sanal olarak evden ulaşabildiğimiz için bir şekilde şanslıyız sanırım! ...
- Math Children Books | MATH FAN
MATH BOOKS FOR YOUNG READERS MATH CHILDREN BOOKS ( K3) MATH BOOKS (Middle School +) MATH BOOKS (4th grade +) A Shapes Book Can you count to a Googol? How do you know what time it is? Fractions in Disguise The Boy Who Loved Math A Hundred Billion Trillion Stars Hidden Figures Nothing stopped Sophie Planetarium Kid Scientists Math Inspectors Of Numbers and Stars Pythagoras Story Series What is the Point of Math Euclid; The Man who invented Geometry Sir Cumference Series Molly and The Mathematical Mysteries Science Comics Series Women in Science 100 Things to Know About Numbers, Computers, and Coding The Element in the Room The Renaisance Thinkers / Inventors / Artists Series Archimedes and His Numbers Timeline of Everything We Have Got Your Number! Solving for M Uncle Petros and Goldbach Conjecture Martin Gardner Puzzles Math, Magic and Mystery The Phantom Tollbooth The adventure of Penrose the mathematical cat The Math Book
- STEAM Library | MATH FAN
S.T.E.A.M Section of the Library Science - Technology - Engineering - Art - Math BOOKS Creating a cozy corner at the library or in the classrooms for STEAM books is to motivate kids is a good idea. To update the books what you have or to create such a corner, there are some easy ways to choose the right books. already There are two leading math books prizes for kids; The first one is MATHICAL This prize is an annual award for fiction and nonfiction books that inspire children of all ages to see math in the world around them. Award-winning books are selected by a nationwide committee of mathematicians like MSRI, NCTM, NCTE, CBC, educators, librarians, early childhood experts, and others. The Books that I can recommend for (everyone) every STEM Teacher to read, to get inspired and learned from... The ROYAL SOCIETY INSIGHT INVESTMENT SCIENCE BOOK PRIZE Another one is The Royal Society is founded on 28 November 1660 as the UK's National Academy of Sciences with the motto of "Nullius in Verba", is Latin for " . Take nobody's word for it The Royal Society awards two prizes each year for the best books communicating science to non-specialists and young people. So by checking the website of the Royal Society, you can get recommendations for the kids an for yourself. You can even see the shortlisted books belong to the previous years. Royal Society Insight Investment Science Book Prize. and Young People's Book Prize Books for Young Show the list Readers S.T.E.A.M MOVIES SECTION OF THE STEAM LIBRARY S.T.E.A.M related inspiring Movies and Documentaries
- Math at Home | MATH FAN
HANDS-ON MATH AT HOME Day 1: Roman Arch Bridges Grab all the cushions, books or Jenga blocks at home and try to build an arch bridge. The forces of a Roman arch so strong that arches can stand without any glue or other adhesive holding them together. Try it for yourself! How it works: Its semicircular structure elegantly distributes compression through its entire form and diverts weight onto its two legs, the components of the bridge that directly take on pressure. Blueprint of the Arch Bridge Home Made Arch Bridge Roman Bridge, Ponte da Vila Formosa, Portugal 1/3 Image attributions: https://www.ancient.eu/image/4407/roman-bridge-ponte-da-vila-formosa-portugal/ https://www.thisiscarpentry.com/2012/01/06/circular-based-arches-part-1/ https://www.thelistlab.net/blog/how-to-make-a-book-arch Resources: https://kids.nationalgeographic.com/explore/books/make-this/roman-ice-arch/ Day 2: Leonardo Da Vinci’s Famous Self-Supporting Bridge Do you have popsicles at home? I did not try with toothpicks or q-tips, but I think that they may also work. Other than those, “Patience” will be the main thing you will need. Leonardo Da Vinci’s Self-Supporting Bridge is also known as the emergency bridge. No nails, screws, rope, glues, notches, or other fasteners are holding the bridge in place. You can also watch the step by step but first I suggest you try by looking at the image below. instruction video How it works: You will be weaving the sticks together so that the tension between the sticks keeps the bridge together and lifts it off of the ground. You may also watch the on YouTube how a father and son build the bridge at their backyards to motivate yourself to keep going :) video 1/4 Image Attributions: https://www.core77.com/posts/65043/Leonardo-da-Vincis-Ingenious-Design-for-a-Self-Supporting-Bridge Resources: https://thekidshouldseethis.com/post/how-to-make-leonardo-da-vincis-self-supporting-arch-bridge https://www.instructables.com/id/Da-Vinci-Popsicle-Stick-Bridge/ Day 3: Cylindrical Mirror and Anamorphic Art The original is the usage of mirroring paper, but nowadays unwrinkled aluminum foil can be used as well (But because the images are fuzzier, the observations may not be as clear.) And a soda or coke can, or any cylindrical object that you can cover with the aluminum foil is ok. After you create the cylindrical mirror, you may either color an already distorted image (1) or print the polar grid (2) below and create your own anamorphic art. How it works: Making anamorphic drawings involves mechanically distorting an image by transferring the image from the square grid (the original image) onto a polar grid (distorted grid. It is a mapping, or a correspondence, between a cartesian set of coordinates, and a polar set of coordinates. Place your cylindrical mirror on the circle and look into the mirror to see the image restored. Resources: https://anamorphicart.wordpress.com/2010/04/21/cylindrical-mirror-anamorphoses/ https://raft.net/wp-content/uploads/2019/03/278-Anamorphic-Art.pdf https://makezine.com/projects/draw-distorted-pictures/ Coloring a Distorted Image (1) You may color the distorted image below from my sure that the cylindrical shape you will find at home matches the circle at. The center of the paper. makezine.com, in that case, : Link for already distorted Makey Bot image Polar Grid Template (2) You can also use this polar grid by printing to make your own drawings; Link for the Polar Grid Day 4: Pi at home You can do lots of different pi activities at home. I want to list a few very popular ones. Pi – skyline: All you need is paper, ruler and crayons, create black bars at the lengths of the digits of Pi and create the skyline for Pi-York, Pi-ris, Pi-lan, Pi-chester …. Building Pi-City with Lego: Instead of coloring the digits of Pi, you can use the lego pieces to actually build your Pi-city Pi – bracelet: If you have the toolkit, all you need is to give each color a number like; 1 -pink, 2-blue, 3-green, 4-red .. And you can start forming your pi bracelet. Pi Art in a Circle: Simply divide a circle into 10 equal intervals label them from 0 to 9 if possible each with different colors. Start drawing lines from 3 to 1, 1 to 4, 4 to 1 and go on … Use the same color for the segment with your starting point for each of the drawings … Pi – Dart Game If You have Dart Board at home by throwing dart, you can calculate Pi. Here all you can do is watching this video. Before I forget, Pi number; 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... Image Attributions and for more information please visit; https://www.whatdowedoallday.com/ http://www.pinkstripeysocks.com/2014/03/pi-day-activity-make-pi-day-bracelets.html Day 5: String Art Another paper, pencil and ruler only activity. But this is to create your art on the paper. If you want to create some 3d art, you can always use a corkboard, pins, and some string. Even if you have the necessary materials for 3d art, I recommend you start with paper and pencil first. Draw a big “L” shape on a paper and mark the numbers with equal intervals till 15. Or you can use the templates below. Again, all you are gonna do is drawing straight lines with a ruler to connect the points such as the; The first point on y-axis goes to the last point on x-axis The second point on y-axis goes to the second-last point on x-axis … Spoiler Alert: When you have finished you’ll see that you have created a curve by using straight lines. You can extend your initial drawing by converting your L shape to a “+” plus sign Then you can try a 60 angle “<” as your initial figure and complete it to a hexagon by connecting 6 of them from their corners. (Here you can use less number of points on the lines..) String art is a topic with no limits if you feel like you are interested, make sure you’ll make an internet search. Have math fun... 1/9 Links for the String Art Templates: L Shape + Shape Square Shape Octagon 60 degrees Hexagon 1 Hexagon 2 Day 6: Vedic Worms Fill in the multiplication table grid and reduce double-digit numbers to a single digit by adding the digit of the products. Example: If 9×9=81, add the numbers in the sum (8+1), and put the sum of 9 in the square. If the new sum is also double-digit, add those numbers. Example: 7×8=56; 5+6=11; 1+1=2. Place the number 2 in that square. We are going to use this number sequences to create the Vedic Worms which are also spirolaterals Spirolaterals are geometrical figures formed by the repetition of a simple rule. The pattern is formed by drawing line segments of a certain length from a number sequence with a fixed angle and a direction. Although the spirolaterals can be created with any number sequence, we will use the Vedic Squares we have created. That’s why they are also called “VEDIC WORMS”. Start with a row of numbers you choose. (1,2,3,4,5,6,7,8,9) These numbers will determine the length of each ‘step’ of the ‘spirolateral’. 2. Choose a direction; clockwise (CW) or counterclockwise (CCW) 3. Choose a grid type to draw on (In fact here, you are choosing the angle of your movement) Square Grid (90 degrees ) Isometric Grid (60 degrees ) Hexagonal Grid (120 degrees ) … 4. Now start drawing spirals through your list. For example, if we choose CW direction on a square grid with the first row of numbers, It means 1 step up, 2 steps right, 3 steps down and 4 steps left then repeat like 5 steps up, 6 steps right, 7 steps down and 8 steps left and 9 steps up to complete your drawing. Please check about the Vedic Squares, Worms and The Spirolaterals for the necessary materials.. t he post Day 7: Two (Hinged) Mirrors and Shapes If you have two small mirrors at home, it means you are ready for this activity. I had two hinged mirrors that I got from Amazon recently, but any mirrors like some of the foldable vanity mirrors or small Ikea ones will do. In addition to the pair of mirrors, any shapes, tangram pieces, lego pieces, different shaped toys can be used for this activity. 1. Angles, Reflection, Tessellation: Tessellation is covering a surface with a shape(s) without any gaps or overlaps. You can create your tessellation by using lego pieces, shapes anything you can create your design. Then, use the mirrors to enlarge your design: Arrange the mirrors as a straight line (180) to double your design! Arrange the mirrors with a 120-degree angle in between to triple your design! Arrange the mirrors with a 90-degree angle in between to ? your design! Arrange the mirrors with a 60-degree angle in between to ? your design! 2. Lego Pieces, Other Half, Polygons Create a car, a space ship, a dragon whatever you like but only half of it, then use the mirror the create the other half. You can do the same by holding the mirrors with different angles to enlarge your designs. Now use a thin, long lego piece, or any toy that you can use as a line segment. Arrange your mirrors with a 120 degrees, put the lego piece in between, what is the name of the polygon you have created? Try the other angles (You can measure the angles with a protractor) what kind of polygons you can create? What if you want to form a polygon with 12 sides (dodecagon), how are you going to arrange the mirrors? You can repeat the same activity by drawing a line segment on a paper and putting the mirrors on it by creating different angles between them. 3. Fractions and Creativity Let's try something else, if you have two identical triangles like tangram pieces ( if don't simply draw, color and cut two identical triangles from paper) arrange them in all the possible ways to create a square by using the mirrors? Which angle you need to use to create the square? How many different designs can you make? What fraction of your design is purple? What about your initial shape? Are those fractions equal? How many different ways you can divide a square into halves? If you want to create a snowflake with a shortcut, what would be the angles between your mirrors? If you want to draw an octopus by drawing only one of its arms, then which angle you need to use? BY USING TWO MIRRORS, YOU CAN LEARN ABOUT; ANGLES POLYGONS TESSELLATION SYMMETRY REFLECTION FRACTIONS ..