This was created using *Stella 4d*, a program you can buy, or try for free, at http://www.software3d.com/Stella.php.

This would give Euler’s identity the value of:

e^iπ + 1 = 1.3509198071784109

So it’s about equal to this number, which is frankly pretty unremarkable, but if this were true it would have a lot of implications for quantum mechanics so if you have any thoughts please leave them in the comments.

]]>For many challenging questions to really get you thinking, try the brilliant

You can search by Station, suppose you want to practise your Algebra – try the **Thinking About Algebra Station **for example where you will find everything from **Equation Sudoku** to some challenging **Surd manipulations.**

On the subject of surds – try **Scary Sum**!

If you create a (free) account you can save and categorise your favourite resources.

There are **many Underground Mathematics Resource Types**. Try the **Review Questions** for example, which in the words of the Underground Maths Team:

These are questions designed to test students’ understanding of one or more topics and to exercise their problem-solving skills. In many cases they can also be used as a classroom resource to help teach concepts and methods. They are mostly drawn from past examination questions and have been chosen as ones that are interesting in nature and require non-routine thinking. The hints and solutions are designed to explain the reasoning and highlight connections as well as giving the answer. In many cases, alternative methods or solutions are presented.

Note the various question types available; these include very challenging questions for students age 16+.

The **Oxford MAT collection** includes an extensive selection of Multiple Choice Questions.

**O/AO-level questions** are included. These questions provide excellent challenge for sudents aspiring to the top grades for examinations taken at age 15-16 and beyond..

**Can we fully factorise x ^{4}+4y^{4}?**Starts with a Show that….

We could get very sophisticated and look at those quadratic factors too; useful for those studying the Level 2 Further Mathematics Qualification.

**Can we simplify these algebraic fractions?
**Review algebraic fractions

**Can we simplify these simultaneous equations of degree 1 and 2?
**Solve simultaneous equations. We will need to factorise a quadratic in this problem with a coefficient which is not 1 for the square term. My students and I are fans of the

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The Online Encyclopedia of Integer Sequences. Because.

(Visit the page of its parent, The OEIS Foundation— movies, posters, and more!)

* Carl Friedrich Gauss

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**As we count ’em up,** we might send starry birthday greeting to Erasmus Reinhold; he was born on this date in 1511. A mathematician and astronomer, Reinhold was considered to be the most influential astronomical pedagogue of his generation. Today, he is probably best known for his carefully calculated set of planetary tables– the first– applying Copernican theory, published in 1551.

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Elsewhere you can find plenty of information about the career and achievements of Dr. Liskov, but this post just took the fruition of an afternoon of thinking about the History of Computer Science by an important figure in its most relevant developments – with the added feature of being a rare female protagonist in those developments, a point rightly mentioned in the talk -; her role in the development of Object-oriented programming, and the current interesting issues about the relationship between correct and proper Software Engineering and the resurgent field of Artificial Intelligence.

Without further explanations, please enjoy this video:

]]>*Q.3) If N=54535352…… 987654321 , find the remainder when N is divided by 9.*

*Q.4) If no. of factors of (2^10) *(3^n) *(10^6) is 1309, then find the value of n.*

*Q.5) A two-digit number is four times the sum of its digits and two times the product of its digits. How many such numbers are possible? *

As I begin to explore ways to build an awareness of the world I always think about what are the tools available to use to help capture and enhance the amazing discoveries we are making inside Kindergarten. So this year I am using Twitter, Kidblog, Padlet, AirServer,Blogger, Skype and GoogleHangout. These platforms, applications and tools all offer opportunities to share what my students are curious about and also capture their ideas that we share globally. I think the sharing no matter what platform you use is a wonderful way to demonstrate explicitly what you are exploring and then your students have opportunities to see who in the world is interested in their learning. This is so powerful for children because they are inspired and excited to see where in the world they are having an impact and who is also interested in what they are learning.

Recently we used **Skype** as a way to learn about the life of bees with an expert. My student intern created a unit of study about bees. She use Skype as 1 way for the children to ask questions and share their learning.

**Twitter** is on all day because you never know when you might want to share an idea. This way my students get to connect with others who are sharing what they are learning. When we “tweet” we are demonstrating and modeling “how to” have a conversation on line. Look here and here to read posts I have had published in regards to using Twitter.

**Padlet **is a fun and easy way to ask others globally for ideas and to share. This tool is like a sticky note so when I explain and share with my students I have something explicit to connect this with. My students were able to see what a pumpkin looked like in Germany and Australia recently via this tool. A great opportunity to make connections and see what is different, the same and wonder about what all of these plants need to grow no matter where in the world they are. Look here at ours.

**Kidblog** is a wonderful way to explore ‘how to” write with an audience in mind. This way my students begin to experience that a larger audience will be reading and looking at there posts, so they become excited and focused on doing their best. **Kidblog **also has a map of the world like we have on our classroom wall and our class blog so we have many opportunities to connect and see who in the world is interested in our ideas. Look here, here and here to read posts Kidblog has published.

**Blogger** is another way to enhance and share your learning and that of your students. Through our own ability to be transparent and share our voice, we offer invitations to share our students’ voices too. The families of my students love the class blog because it offers them an opportunity to have face to face conversations with their child about there day inside Kindergarten. This is important because I want the learning that takes place inside my classroom to be outside as well. This is whereby students begin to develop an awareness of the world and how many wonderful things we have to share. Look here to see a post that was published on blogging and here to read about the positive effects of blogging. Why I blog…..

**GoogleHangout **is an easy and fun way for parents to read a story to the class as well as share with other children globally about a topic of interest. Read Across America is a great opportunity for teachers to connect with others globally via a great book!

This year many parents have been emailing me photos of their child making connections with ideas we are exploring inside Kindergarten. This has been a wonderful way for my students to share their connection with the class and facilitate a discussion. This type of opportunity also gives all my students inspiration and raises their self confidence. They matter and what is important to them matters too.

In the photos you might notice a child seeing a rhombus created with light in her home. This is one of the attributes we are exploring in Kindergarten. Another child sharing a climb he made also noticing a circular shape on top of a mountain. He shared what he saw as well while on top of a mountain. This is another opportunity for me to weave in the idea of perspective. Another child creating pizza. We had an opportunity to talk about straight lines and curvy lines because of what this child created. We were also able to justify our thinking because we made connections with other things that are triangular.

Interacting with the world has opened my eyes to what is possible to explore even with 5 and 6 year old children. We are able to enrich our understanding of the world through our ideas and sharing our perspectives with others. Then we make connections with what we see others doing and we begin to question which is where we think critically and deeply about what we are exploring.

]]>Here is today’s homework.

With regards,

Preeti Lashkari

]]>Here is today’s homework.

With regards,

Charu Soni

]]>How exactly? Because fibonacci is present in all forms of nature, everywhere you look, is a fibonacci pattern.

Source: Quora

]]>The great icosahedron, one of the Kepler-Poinsot solids, is hidden from view at the center of this cluster. Each of its faces is augmented with a Platonic icosahedron, producing what you see here. *Stella 4d* is the software I used; more information about that program may be found here.

*All closed surfaces can be produced by gluing the sides of some polygon and all even-sided polygons (2n-gons) can be glued to make different manifolds.*

*Conversely, a closed surface with non-zero classes can be cut into a 2n-gon.*

Two interesting cases of this are:

- Gluing opposite sides of a hexagon produces a torus .
- Gluing opposite sides of an octagon produces a surface with two holes, topologically equivalent to a torus with two holes.

I had trouble visualizing this on a piece of paper, so I found two videos which are fascinating and instructive, respectively.

**The two-torus from a hexagon**

**The genus-2 Riemann surface from an octagon**

I would like to figure out how one can make such animations, and generalizations of these, using Mathematica or Sagemath.

There are a bunch of other very cool examples on the Youtube channels of these users. Kudos to them for making such instructive videos!

PS – I see that $\LaTeX$ on WordPress has become (or is still?) very sloppy! :(

]]>May 2016-July 2016

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**Contact Barbara Boe**

Kurt Friedrich Gödel was a mathematician, logician and philosopher and widely regarded as one of the most important figures in mathematical logic (and certainly the most important contemporary). At the tender age of 25, Gödel was able to prove his incompleteness theorems, the results of which have far reaching impact. I really should try and pack a little more into my years. His theorems can be summarized as;

- Within any system where enough arithmetic can be carried out there are statements written in the language of this system that can’t be proved or disproved.
- A system cannot prove that the system itself is consistent, assuming it is in the first place (which we could never prove!).

Roughly speaking a “system” means a set of rules within which we can create new theorems. We generally call these rules axioms; they are the truths of the system if you like. We then have operations we may perform within our system, which coupled with the axioms allow us to determine for any formula (in the language of the system) if it is a proper derivation or proof *within the system*. The natural numbers are a very basic example, coupled with the arithmetic operators we know so well. If the above definition didn’t come naturally it might be worth a reread; it is quite important. Now a system can be described as complete if for any statement we generate within the system we can prove/derive the statement (within the system). A system is consistent if, for every provable statement I cannot also correctly prove the contra. To use an example; if I can prove I am a man, for the system to be consistent I better not be able to also prove I am a woman; which, to my knowledge I cannot.

So what does all of this mean? It is really important to understand that a statement being false is not analogous with a disprovable statement. Although not a perfect example, consider the idea that no number of white swans may prove the statement that all swans are white – all swans might be white, but even with a large army of white swans I am nothing more than a man with lots of swans. In order to get your head around this we think about computer programming; if you read about Gödel’s theorems this is what you will often see cited.

We use a “perfect” computer where we don’t worry about petty things like processors, electricity and alike. Just one big super computer that exists (in your mind). The computer can perform the usual arithmetic operations of addition, subtraction, division and multiplication. You can program your computer to perform tasks, you really can pick anything you like it’s your computer. You might be interested in knowing if a value is larger than 100 – well tell the computer to subtract 100 from the value and if the result is 0 or less display false. We can get more sophisticated and program our wonderful computer to determine primes (a sequence of divisions), Fibonacci numbers or square numbers. We can do *whatever* we like with our supercomputer, right? Wrong.

As a direct consequence of Gödel’s theorems, there are statements (only involving arithmetic, we aren’t asking the meaning of life), correctly inputted into our not so super computer which it cannot prove or disprove. The computer genuinely cannot decide if the outcome is true or false; nothing worse than an indecisive computer. To make things spookier, say you introduce a new axiom to the computer, a cheap work around to say that the bothersome statement is either true or false (it may well be intuitive) things don’t get better. You would either generate more statements the computer cannot decide between (1) or you would create results that contradict the pre-improved computer which are valid in the new post-improved computer – inconsistent (2). When this whole math-bomb dropped people hoped it was just a quirk of the system as defined rather than the reality of systems we know and love. Alan Turing, among others showed that this is reality. An incomplete or inconsistent mathematical meltdown keeping logicians up at night pondering all they once knew.

The theory has been used in a huge number of areas; including as an attempt to prove the existence of God I know some of my readers will be delighted to know! But alas, that is not the direction I intend to take for the swansong; instead I delve inside your mind which is much more interesting. I exaggerate a little, we will all need to journey deep inside our own minds. If we consider our brains as essentially machines (your choice), then do Gödel’s theorems apply? We hope – dearly – that our brains are consistent (although I do sometimes find myself questioning this watching political commentary today). So we arrive at the conclusion, based on the above, that there are basic statements with arithmetic operations we cannot prove or disprove within our own brain – which would be the strongest line of evidence that we cannot ever construct or program a human mind. Rodger Penrose is a firm believer of this, who controversially states that a human mind is capable of *knowing* the truth about these Gödel-disprovable statements; this form of intelligence is never able to be computed. More on this in the future, but I should tell you this has been met with some severe criticism; centrally that we cannot know a human brain is consistent without much more understanding of the inner workings, especially due to the number of mistakes we make we could very plausibly be neurologically inconsistent, (and perhaps some more than others!).

That aside, Gödel’s theorems raise some profound questions around the foundations of mathematics and the nature of our brains. The mathematical proof is quite dense and I do not intend to outline it; if you are interested I actually find Wikipedia the most usable but please let me know if you have more user-friendly sources. The language of logic is a difficult beast to tame.

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