What's all this, then?
https://numbersandshapes.net/
Recent content on What's all this, then?Hugo -- gohugo.ioen-usSat, 18 Jan 2020 00:00:00 +1100About this blog
https://numbersandshapes.net/pages/about/
Mon, 01 Jan 0001 00:00:00 +0000https://numbersandshapes.net/pages/about/This is the newest version of my “Numbers and Shapes” blog, containing material on elementary mathematics, mathematics education, software, and anything else that takes my fancy.
It is created in Hugo, a static blog engine which is blindingly fast, and works by simply serving up static html files (along with any necessary javascript). There is no database, and far less to fiddle with than with wordpress. Because it is static there is less opportunity for user input (such as comments) but that’s not an issue for me, as about 99.Pictures
https://numbersandshapes.net/pages/pictures/
Mon, 01 Jan 0001 00:00:00 +0000https://numbersandshapes.net/pages/pictures/This page will contain various photos and snapshots.
New Zealand, 2018: First, from a holiday taken by my wife and me through the South Island of New Zealand early in 2018:
New Zealand 2018
Taiwan, 2017: From a conference in Taiwan (Taoyuan City mainly, but also in Taipei and including a visit to Keelung City) in December 2017:
Taiwan 2017
Shanghai, December 2015: I spent several wonderful days here en route to a conference.Enumerating the rationals
https://numbersandshapes.net/post/enumerating_the_rationals/
Sat, 18 Jan 2020 00:00:00 +1100https://numbersandshapes.net/post/enumerating_the_rationals/The rational numbers are well known to be countable, and one standard method of counting them is to put the positive rationals into an infinite matrix \(M=m_{ij}\), where \(m_{ij}=i/j\) so that you end up with something that looks like this:
\[ \left[\begin{array}{ccccc} \frac{1}{1}&\frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\dots\\[1ex] \frac{2}{1}&\frac{2}{2}&\frac{2}{3}&\frac{2}{4}&\dots\\[1ex] \frac{3}{1}&\frac{3}{2}&\frac{3}{3}&\frac{3}{4}&\dots\\[1ex] \frac{4}{1}&\frac{4}{2}&\frac{4}{3}&\frac{4}{4}&\dots\\[1ex] \vdots&\vdots&\vdots&\vdots&\ddots \end{array}\right] \]
It is clear that not only will each positive rational appear somewhere in this matrix, but its value will appear an infinite number of times.Fitting the SIR model of disease to data in Julia
https://numbersandshapes.net/post/fitting_sir_to_data_in_julia/
Wed, 15 Jan 2020 00:00:00 +1100https://numbersandshapes.net/post/fitting_sir_to_data_in_julia/A few posts ago I showed how to do this in Python. Now it’s Julia’s turn. The data is the same: spread of influenza in a British boarding school with a population of 762. This was reported in the British Medical Journal on March 4, 1978, and you can read the original short article here.
As before we use the SIR model, with equations
\begin{align*} \frac{dS}{dt}&=-\frac{\beta IS}{N}\\
\frac{dI}{dt}&=\frac{\beta IS}{N}-\gamma I\\The Butera-Pernici algorithm (2)
https://numbersandshapes.net/post/the_butera_pernici_algorithm_2/
Mon, 06 Jan 2020 00:00:00 +1100https://numbersandshapes.net/post/the_butera_pernici_algorithm_2/The purpose of this post will be to see if we can implement the algorithm in Julia, and thus leverage Julia’s very fast execution time.
We are working with polynomials defined on nilpotent variables, which means that the degree of any generator in a polynomial term will be 0 or 1. Assume that our generators are indexed from zero: \(x_0,x_1,\ldots,x_{n-1}\), then any term in a polynomial will have the form \[ cx_{i_1}x_{i_2}\cdots x_{i_k} \] where \(\{x_{i_1}, x_{i_2},\ldots, x_{i_k}\}\subseteq\{0,1,2,\ldots,n-1\}\).The Butera-Pernici algorithm (1)
https://numbersandshapes.net/post/the_butera_pernici_algorithm_1/
Sat, 04 Jan 2020 00:00:00 +1100https://numbersandshapes.net/post/the_butera_pernici_algorithm_1/Introduction We know that there is no general sub-exponential algorithm for computing the permanent of a square matrix. But we may very reasonably ask – might there be a faster, possibly even polynomial-time algorithm, for some specific classes of matrices? For example, a sparse matrix will have most terms of the permanent zero – can this be somehow leveraged for a better algorithm?
The answer seems to be a qualified “yes”.The size of the universe
https://numbersandshapes.net/post/the_size_of_the_universe/
Thu, 02 Jan 2020 00:00:00 +1100https://numbersandshapes.net/post/the_size_of_the_universe/As a first blog post for 2020, I’m dusting off one from my previous blog, which I’ve edited only slightly.
I’ve been looking up at the sky at night recently, and thinking about the sizes of things. Now it’s all very well to say something is for example a million kilometres away; that’s just a number, and as far as the real numbers go, a pretty small one (all finite numbers are “small”).Permanents and Ryser's algorithm
https://numbersandshapes.net/post/permanents_and_rysers_algorithm/
Sun, 22 Dec 2019 00:00:00 +1100https://numbersandshapes.net/post/permanents_and_rysers_algorithm/As I discussed in my last blog post, the permanent of an \(n\times n\) matrix \(M=m_{ij}\) is defined as \[ \text{per}(M)=\sum_{\sigma\in S_n}\prod_{i=1}^nm_{i,\sigma(i)} \] where the sum is taken over all permutations of the \(n\) numbers \(1,2,\ldots,n\). It differs from the better known determinant in having no sign changes. For example: \[ \text{per} \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} =aei+afh+bfg+bdi+cdi+ceg. \] By comparison, here is the determinant: \[ \text{det} \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} =aei - afh + bfg - bdi + cdi - ceg.Speeds of Julia and Python
https://numbersandshapes.net/post/speeds_of_julia_and_python/
Thu, 19 Dec 2019 00:00:00 +1100https://numbersandshapes.net/post/speeds_of_julia_and_python/Introduction Python is of course one of the world’s currently most popular languages, and there are plenty of statistics to show it. Of all languages in current use, Python is one of the oldest (in the very quick time-scale of programming languages) dating from 1990 - only C and its variants are older. However, it seems to keep its eternal youth by being re-invented, and by its constantly increasing libraries.Poles of inaccessibility
https://numbersandshapes.net/post/poles_of_inaccessibility/
Sun, 08 Dec 2019 00:00:00 +1100https://numbersandshapes.net/post/poles_of_inaccessibility/Just recently there was a news item about a solo explorer being the first Australian to reach the Antarctic “Pole of Inaccessibility”. Such a Pole is usually defined as that place on a continent that is furthest from the sea. The South Pole is about 1300km from the nearest open sea, and can be reached by specially fitted aircraft, or by tractors and sleds along the 1600km “South Pole Highway” from McMurdo Base.An interesting sum
https://numbersandshapes.net/post/an_interesting_sum/
Mon, 02 Dec 2019 00:00:00 +1100https://numbersandshapes.net/post/an_interesting_sum/I am not an analyst, so I find the sums of infinite series quite mysterious. For example, here are three. The first one is the value of \(\zeta(2)\), very well known, sometimes called the “Basel Problem” and first determined by (of course) Euler: \[ \sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}. \] Second, subtracting one from the denominator: \[ \sum_{n=2}^\infty\frac{1}{n^2-1}=\frac{3}{4} \] This sum is easily demonstrated by partial fractions: \[ \frac{1}{n^2-1}=\frac{1}{2}\left(\frac{1}{n-1}-\frac{1}{n+1}\right) \] and so the series can be expanded as: \[ \frac{1}{2}\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{5}\cdots\right) \] This is a telescoping series in which every term in the brackets is cancelled except for \(1+1/ 2\), which produces the sum immediately.Runge's phenomenon in Geogebra
https://numbersandshapes.net/post/runges_phenomenon_in_geogebra/
Sun, 15 Sep 2019 00:00:00 +1000https://numbersandshapes.net/post/runges_phenomenon_in_geogebra/Runge’s phenomenon says roughly that a polynomial through equally spaced points over an interval will wobble a lot near the ends. Runge demonstrated this by fitting polynomials through equally spaced point in the interval \([-1,1]\) on the function \[ \frac{1}{1+25x^2} \] and this function is now known as “Runge’s function”.
It turns out that Geogebra can illustrate this extremely well.
Equally spaced vertices Either open up your local version of Geogebra, or go to http://geogebra.Fitting the SIR model of disease to data in Python
https://numbersandshapes.net/post/fitting_sir_to_data_in_python/
Fri, 09 Aug 2019 00:00:00 +1000https://numbersandshapes.net/post/fitting_sir_to_data_in_python/Introduction and the problem The SIR model for spread of disease was first proposed in 1927 in a collection of three articles in the Proceedings of the Royal Society by Anderson Gray McKendrick and William Ogilvy Kermack; the resulting theory is known as Kermack–McKendrick theory; now considered a subclass of a more general theory known as compartmental models in epidemiology. The three original articles were republished in 1991, in a special issue of the Bulletin of Mathematical Biology.Mapping voting gains between elections
https://numbersandshapes.net/post/mapping_voting_gains_between_elections/
Sun, 21 Jul 2019 00:00:00 +1000https://numbersandshapes.net/post/mapping_voting_gains_between_elections/So this goes back quite some time to the recent Australian Federal election on May 18. In my own electorate (known formally as a “Division”) of Cooper, the Greens, who until recently had been showing signs of winning the seat, were pretty well trounced by Labor.
Some background asides First, “Labor” as in “Australian Labor Party” is spelled the American way; that is, without a “u”, even though “labour” meaning work, is so spelled in Australian English.Educational disciplines: size against market growth
https://numbersandshapes.net/post/educational_disciplines/
Sat, 15 Jun 2019 00:00:00 +1000https://numbersandshapes.net/post/educational_disciplines/Here is an interactive version of this diagram:
(click on the image to show a larger version.)Tschirnhausen transformations and the quartic
https://numbersandshapes.net/post/tschirnhausens_transformations_quartic/
Sun, 05 May 2019 00:00:00 +1000https://numbersandshapes.net/post/tschirnhausens_transformations_quartic/Here we show how a Tschirnhausen transformation can be used to solve a quartic equation. The steps are:
Ensure the quartic is missing the cubic term, and its initial coefficient is 1. We can do this by first dividing by the initial coefficient to obtain an equation \[ x^4+b_3x^3+b_2x^2+b_1x+b_0=0 \] and then replace the variable \(x\) with \(y=x-b_3/ 4\). This will produce a monic quartic equation missing the cubic term.Tschirnhausen's solution of the cubic
https://numbersandshapes.net/post/tschirnhausens_solution_of_the_cubic/
Sun, 05 May 2019 00:00:00 +1000https://numbersandshapes.net/post/tschirnhausens_solution_of_the_cubic/A general cubic polynomial has the form \[ ax^3+bx^2+cx+d \] but a general cubic equation can have the form \[ x^3+ax^2+bx+c=0. \] We can always divide through by the coefficient of \(x^3\) (assuming it to be non-zero) to obtain a monic equation; that is, with leading coefficient of 1. We can now remove the \(x^2\) term by replacing \(x\) with \(y-a/3\): \[ \left(y-\frac{a}{3}\right)^{\negmedspace 3}+a\left(y-\frac{a}{3}\right)^{\negmedspace 2} +b\left(y-\frac{a}{3}\right)+c=0. \] Expanding and simplifying produces \[ y^3+\left(b-\frac{a^2}{3}\right)y+\frac{2}{27}a^3-\frac{1}{3}ab+c=0.Colonial massacres, 1794 to 1928
https://numbersandshapes.net/post/colonial_massacres/
Mon, 28 Jan 2019 00:00:00 +1100https://numbersandshapes.net/post/colonial_massacres/The date January 26 is one of immense current debate in Australia. Officially it’s the date of Australia Day, which supposedly celebrates the founding of Australia. To Aboriginal peoples it is a day of deep mourning and sadness, as the date commemorates over two centuries of oppression, bloodshed, and dispossession. To them and their many supporters, January 26 is Invasion Day.
The date commemorates the landing in 1788 of Arthur Phillip, in charge of the First Fleet and the first Governor of the colony of New South Wales.Vote counting in the Australian Senate
https://numbersandshapes.net/post/vote_counting_in_australian_senate/
Tue, 22 Jan 2019 00:00:00 +1100https://numbersandshapes.net/post/vote_counting_in_australian_senate/Recently we have seen senators behaving in ways that seem stupid, or contrary to accepted public opinion. And then people will start jumping up and down and complaining that such a senator only got a tiny number of first preference votes. One commentator said that one senator, with 19 first preference votes, “couldn’t muster more than 19 members of his extended family to vote for him”. This displays an ignorance of how senate counting works.Concert review: Lixsania and the Labyrinth
https://numbersandshapes.net/post/lixsania_and_labyrinth/
Sat, 10 Nov 2018 00:00:00 +1100https://numbersandshapes.net/post/lixsania_and_labyrinth/This evening I saw the Australia Brandenburg Orchestra with guest soloist Lixsania Fernandez, a virtuoso player of the viola da gamba, from Cuba. (Although she studied, and now lives, in Spain.) Lixsania is quite amazing: tall, statuesque, quite absurdly beautiful, and plays with a technique that encompasses the wildest of baroque extravagances as well as the most delicate and refined tenderness.
The trouble with the viol, being a fairly soft instrument, is that it’s not well suited to a large concert hall.Linear programming in Python (2)
https://numbersandshapes.net/post/linear_programming_in_python_2/
Tue, 30 Oct 2018 00:00:00 +1100https://numbersandshapes.net/post/linear_programming_in_python_2/Here’s an example of a transportation problem, with information given as a table:
.transport table { width: 80%; table-layout: fixed;} th, td { text-align: center;} th { background: #DCDCDC;} Demands 300 360 280 340 220 750 100 150 200 140 35 Supplies 400 50 70 80 65 80 350 40 90 100 150 130 This is an example of a balanced, non-degenerate transportation problem.Linear programming in Python
https://numbersandshapes.net/post/linear_programming_in_python/
Sun, 28 Oct 2018 00:00:00 +1100https://numbersandshapes.net/post/linear_programming_in_python/For my elementary linear programming subject, the students (who are all pre-service teachers) use Excel and its Solver as the computational tool of choice. We do this for several reasons: Excel is software with which they’re likely to have had some experience, also it’s used in schools; it also means we don’t have to spend time and mental energy getting to grips with new and unfamiliar software. And indeed the mandated curriculum includes computer exploration, using either Excel Solver, or the Wolfram Alpha Linear Programming widget.A test of OpenJSCAD
https://numbersandshapes.net/post/test_of_openjscad/
Sat, 15 Sep 2018 00:00:00 +1000https://numbersandshapes.net/post/test_of_openjscad/Here’s an example of a coloured tetrahedron:
hello The power of two irrational numbers being rational
https://numbersandshapes.net/post/powers_of_irrationals/
Sat, 15 Sep 2018 00:00:00 +1000https://numbersandshapes.net/post/powers_of_irrationals/There’s a celebrated elementary result which claims that:
There are irrational numbers \(x\) and \(y\) for which \(x^y\) is rational.
The standard proof goes like this. Now, we know that \(\sqrt{2}\) is irrational, so let’s consider \(r=\sqrt{2}^\sqrt{2}\). Either \(r\) is rational, or it is not. If it is rational, then we set \(x=\sqrt{2}\), \(y=\sqrt{2}\) and we are done. If \(r\) is irrational, then set \(x=r\) and \(y=\sqrt{2}\). This means that \[ x^y=\left(\sqrt{2}^\sqrt{2}\right)^{\sqrt{2}}=\sqrt{2}^2=2 \] which is rational.Wrestling with Docker
https://numbersandshapes.net/post/wrestling_with_docker/
Sat, 15 Sep 2018 00:00:00 +1000https://numbersandshapes.net/post/wrestling_with_docker/For years I have been running a blog and other web apps on a VPS running Ubuntu 14.04 and Apache - a standard LAMP system. However, after experimenting with some apps - temporarily installing them and testing them, only to discard them, the system was becoming a total mess. Worst of all, various MySQL files were ballooning out in size: the ibdata1 file in /var/lib/mysql was coming in at a whopping 37Gb (39568015360 bytes to be more accurate).Householder's methods
https://numbersandshapes.net/post/householders_methods/
Sun, 09 Sep 2018 00:00:00 +1000https://numbersandshapes.net/post/householders_methods/These are a class of root-finding methods; that is, for the numerical solution of a single nonlinear equation, developed by Alston Scott Householder in 1970. They may be considered a generalisation of the well known Newton-Raphson method (also known more simply as Newton’s method) defined by
\[ x\leftarrow x-\frac{f(x)}{f’(x)}. \]
where the equation to be solved is \(f(x)=0\).
From a starting value \(x_0\) a sequence of iterates can be generated byThe Joukowsky Transform
https://numbersandshapes.net/post/joukowsky-transform/
Fri, 24 Aug 2018 00:00:00 +1000https://numbersandshapes.net/post/joukowsky-transform/The Joukowksy Transform is an elegant and simple way to create an airfoil shape.
Let \(C\) be a circle in the complex plane that passes through the point \(z=1\) and encompasses the point \(z=-1\). The transform is defined as
\[ \zeta=z+\frac{1}{z}. \]
We can explore the transform by looking at the circles centred at \((-r,0)\) with \(r<0\) and with radius \(1+r\):
\[ \|z-r\|=1+r \]
or in cartesian coordinates with parameter \(t\):Double Damask
https://numbersandshapes.net/post/double-damask/
Fri, 27 Apr 2018 00:00:00 +1000https://numbersandshapes.net/post/double-damask/This was a comedy sketch initially performed in the revue “Clowns in Clover” which had its first performance at the Adelphi Theatre in London on December 1, 1927. This particular sketch was written by Dion Titheradge and starred the inimitable Cicely Courtneidge as the annoyed customer Mrs Spooner. It has been recorded and is available on many different collections; you can also hear it on youtube.
I have loved this sketch since I first heard it as a teenager on a three record collection called something like “Masters of Comedy”, being a collection of classic sketches.Numbers and Shapes
https://numbersandshapes.net/nshapes/numbers_and_shapes/
Tue, 24 Apr 2018 00:00:00 +1000https://numbersandshapes.net/nshapes/numbers_and_shapes/This is the holding place for my old blog, when I import it from Wordpress. In the meantime, it’s herePost 1
https://numbersandshapes.net/nshapes/post_1/
Tue, 24 Apr 2018 00:00:00 +1000https://numbersandshapes.net/nshapes/post_1/Just to see how it worksPost 2
https://numbersandshapes.net/nshapes/post_2/
Tue, 24 Apr 2018 00:00:00 +1000https://numbersandshapes.net/nshapes/post_2/and another one, for experiment.Graphs of Eggs
https://numbersandshapes.net/post/egg_graphs/
Fri, 20 Apr 2018 00:00:00 +1000https://numbersandshapes.net/post/egg_graphs/I recently came across some nice material on John Cook’s blog about equations that described eggs.
It turns out there are vast number of equations whose graphs are egg-shaped: that is, basically ellipse shape, but with one end “rounder” than the other.
You can see lots at Jürgen Köller’s Mathematische Basteleien page. (Although this blog is mostly in German, there are enough English language pages for monoglots such as me). And plenty of egg equations can be found in the 2dcurves pages.Exploring JSXGraph
https://numbersandshapes.net/post/exploring_jsxgraph/
Sat, 14 Apr 2018 00:00:00 +1000https://numbersandshapes.net/post/exploring_jsxgraph/JSXGraph is a graphics package deveoped in Javascript, and which seems to be tailor-made for a static blog such as this. It consists of only two files: the javascript file itself, and an accompanying css file, which you can download. Alternaively you can simply link to the online files at the Javascript content delivery site cdnjs managed by cloudflare. There are cloudflare servers all over the world - even in my home town of Melbourne, Australia.The trinomial theorem
https://numbersandshapes.net/post/trinomial_theorem/
Thu, 05 Apr 2018 00:00:00 +1000https://numbersandshapes.net/post/trinomial_theorem/When I was teaching the binomial theorem (or, to be more accurate, the binomial expansion) to my long-suffering students, one of them asked me if there was a trinomial theorem. Well, of course there is, although in fact expanding sums of greater than two terms is generally not classed as a theorem described by the number of terms. The general result is
\[ (x_1+x_2+\cdots+x_k)^n=\sum_{a_1+a_2+\cdots+a_k=n} {n\choose a_1,a_2,\ldots,a_k}x_1^{a_1}x_2^{a_2}\cdots x_k^{a_k} \]
so in particular a “trinomial theorem” would bePlaying with Hugo
https://numbersandshapes.net/post/playing_with_hugo/
Tue, 03 Apr 2018 00:00:00 +1000https://numbersandshapes.net/post/playing_with_hugo/I’ve been using wordpress as my blogging platform since I first started, about 10 years ago. (In fact the first post I can find is dated March 30, 2008.) I chose wordpress.com back then because it was (a) free, and (b) supported mathematics through a version (or subset) of LaTeX. As I have used LaTeX extensively for all my writing since the early 1990’s, it’s a standard requirement for me.Python GIS, and election results
https://numbersandshapes.net/post/python_gis/
Sat, 31 Mar 2018 00:00:00 +1100https://numbersandshapes.net/post/python_gis/Election mapping A few weeks ago there was a by-election in my local electorate (known as an electoral division) of Batman here in Australia. I was interested in comparing the results of this election with the previous election two years ago. In this division it’s become a two-horse race: the Greens against the Australian Labor Party. Although Batman had been a solid Labor seat for almost its entire existence - it used to be considered one of the safest Labor seats in the country - over the past decade or so the Greens have been making inroads into this Labor heartland, to the extent that is no longer considered a safe seat.Presentations and the delight of js-reveal
https://numbersandshapes.net/post/presentations_and_js_reveal/
Sun, 11 Mar 2018 00:00:00 +1100https://numbersandshapes.net/post/presentations_and_js_reveal/Presentations are a modern bugbear. Anybody in academia or business, or any professional field really, will have sat through untold hours of presentations. And almost all of them are terrible. Wordy, uninteresting, too many “transition effects”, low information content, you know as well as I do.
Pretty much every speaker reads the words on their slides, as though the audience were illiterate. I went to a talk once which consisted of 60 – yes, sixty – slides of very dense text, and the presenter read through each one.The Vigenere cipher in haskell
https://numbersandshapes.net/post/vigenere_cipher_haskell/
Tue, 23 Jan 2018 00:00:00 +1100https://numbersandshapes.net/post/vigenere_cipher_haskell/Programming the Vigenère cipher is my go-to problem when learning a new language. It’s only ever a few lines of code, but it’s a pleasant way of getting to grips with some of the basics of syntax. For the past few weeks I’ve been wrestling with Haskell, and I’ve now got to the stage where a Vigenère program is in fact pretty easy.
As you know, the Vigenère cipher works using a plaintext and a keyword, which is repeated as often as need be:Analysis of a recent election
https://numbersandshapes.net/post/abalysis_recent_election/
Thu, 07 Dec 2017 00:00:00 +1100https://numbersandshapes.net/post/abalysis_recent_election/On November 18, 2017, a by-election was held in my suburb of Northcote, on account of the death by cancer of the sitting member. It turned into a two-way contest between Labor (who had held the seat since its inception in 1927), and the Greens, who are making big inroads into the inner city. The Greens candidate won, much to Labor’s surprise. As I played a small part in this election, I had some interest in its result.Analysis of a recent election
https://numbersandshapes.net/post/analysis_recent_election/
Thu, 07 Dec 2017 00:00:00 +1100https://numbersandshapes.net/post/analysis_recent_election/On November 18, 2017, a by-election was held in my suburb of Northcote, on account of the death by cancer of the sitting member. It turned into a two-way contest between Labor (who had held the seat since its inception in 1927), and the Greens, who are making big inroads into the inner city. The Greens candidate won, much to Labor’s surprise. As I played a small part in this election, I had some interest in its result.Programmable CAD
https://numbersandshapes.net/post/programmable-cad/
Fri, 24 Nov 2017 00:00:00 +1100https://numbersandshapes.net/post/programmable-cad/Every few years I decide to have a go at using a CAD package for the creation of 3D diagrams and shapes, and every time I give it up. There’s simply too much to learn in terms of creating shapes, moving them about, and so on, and every system seems to have its own ways of doing things. My son (who is an expert in Blender) recommended that I experiment with Tinkercad, and indeed this is probably a pretty easy way of getting started with 3D CAD.