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6.4.2 - 2020-09-21 Bug fix. There is a new type of zone that can be created and is fully accessible from KSP, allowing for dynamic mapping of samples. Sequences - Finding a Rule. To find a missing number in a Sequence, first we must have a Rule. A Sequence is a set of things (usually numbers) that are in order. Each number in the sequence is called a term (or sometimes 'element' or 'member'), read Sequences and Series for a more in-depth discussion. Operators Once introduced to variables and constants, we can begin to operate with them by using operators.What follows is a complete list of operators. At this point, it is likely not necessary to know all of them, but they are all listed here to also serve as reference.

To find a missing number in a Sequence, first we must have a Rule

Sequence

A Sequence is a set of things (usually numbers) that are in order.

Each number in the sequence is called a term (or sometimes 'element' or 'member'), read Sequences and Series for a more in-depth discussion.

Finding Missing Numbers

To find a missing number, first find a Rule behind the Sequence.

Sometimes we can just look at the numbers and see a pattern:

Example: 1, 4, 9, 16, ?

Answer: they are Squares (1²=1, 2²=4, 3²=9, 4²=16, ...)

Rule: x_n = n²

Sequence: 1, 4, 9, 16, 25, 36, 49, ...

Did you see how we wrote that rule using 'x' and 'n' ?

x_n means 'term number n', so term 3 is written x₃

And we can calculate term 3 using:

x₃ = 3² = 9

We can use a Rule to find any term. For example, the 25th term can be found by 'plugging in' 25 wherever n is.

x₂₅ = 25² = 625

How about another example:

Example: 3, 5, 8, 13, 21, ?

After 3 and 5 all the rest are the sum of the two numbers before,

That is 3 + 5 = 8, 5 + 8 = 13 etc, which is part of the Fibonacci Sequence:

3, 5, 8, 13, 21, 34, 55, 89, ...

Which has this Rule:

Rule: x_n = x_n-1 + x_n-2

Now what does x_n-1 mean? It means 'the previous term' as term number n-1 is 1 less than term number n.

And x_n-2 means the term before that one.

Let's try that Rule for the 6th term:

x₆ = x_6-1 + x_6-2

x₆ = x₅ + x₄

So term 6 equals term 5 plus term 4. We already know term 5 is 21 and term 4 is 13, so:

x₆ = 21 + 13 = 34

Many Rules

One of the troubles with finding 'the next number' in a sequence is that mathematics is so powerful we can find more than one Rule that works.

What is the next number in the sequence 1, 2, 4, 7, ?

Here are three solutions (there can be more!):

Solution 1: Add 1, then add 2, 3, 4, ...

So, 1+1=2, 2+2=4, 4+3=7, 7+4=11, etc...

Rule: x_n = n(n-1)/2 + 1

Sequence: 1, 2, 4, 7, 11, 16, 22, ...

(That rule looks a bit complicated, but it works)

Solution 2: After 1 and 2, add the two previous numbers, plus 1:

Rule: x_n = x_n-1 + x_n-2 + 1

Sequence: 1, 2, 4, 7, 12, 20, 33, ...

Solution 3: After 1, 2 and 4, add the three previous numbers

Rule: x_n = x_n-1 + x_n-2 + x_n-3

Sequence: 1, 2, 4, 7, 13, 24, 44, ...

So, we have three perfectly reasonable solutions, and they create totally different sequences.

Which is right? They are all right.

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Simplest Rule

When in doubt choose the simplest rule that makes sense, but also mention that there are other solutions.

Finding Differences

Sometimes it helps to find the differences between each pair of numbers ... this can often reveal an underlying pattern.

Here is a simple case:

The differences are always 2, so we can guess that '2n' is part of the answer.

Let us try 2n:

The last row shows that we are always wrong by 5, so just add 5 and we are done:

Rule: x_n = 2n + 5

OK, we could have worked out '2n+5' by just playing around with the numbers a bit, but we want a systematic way to do it, for when the sequences get more complicated.

Second Differences

In the sequence {1, 2, 4, 7, 11, 16, 22, ...} we need to find the differences ...

... and then find the differences of those (called second differences), like this:

The second differences in this case are 1.

With second differences we multiply by n²2

In our case the difference is 1, so let us try just n²2:

We are close, but seem to be drifting by 0.5, so let us try: n²2 − n2

Wrong by 1 now, so let us add 1:

We did it!

The formula n²2 − n2 + 1 can be simplified to n(n-1)/2 + 1

So by 'trial-and-error' we discovered a rule that works:

Rule: x_n = n(n-1)/2 + 1

Sequence: 1, 2, 4, 7, 11, 16, 22, 29, 37, ...

Other Types of Sequences

Read Sequences and Series to learn about:

And there are also:

And many more!

In truth there are too many types of sequences to mention here, but if there is a special one you would like me to add just let me know.

Verdana is a humanist sans-seriftypeface designed by Matthew Carter for Microsoft Corporation, with hand-hinting done by Thomas Rickner, then at Monotype. Demand for such a typeface was recognized by Virginia Howlett of Microsoft's typography group and commissioned by Steve Ballmer.^[1][2] The name 'Verdana' is based on verdant (something green), and Ana (the name of Howlett's eldest daughter).^[3]

Bearing similarities to humanist sans-serif typefaces such as Frutiger, Verdana was designed to be readable at small sizes on the low-resolution computer screens of the period.^[4] Like many designs of this type, Verdana has a large x-height (tall lower-case characters), with wider proportions and looser letter-spacing than on print-orientated designs like Helvetica. The counters and apertures are wide, to keep strokes clearly separate from one another, and similarly-shaped letters are designed to appear clearly different to increase legibility for body text. The bold weight is thicker than would be normal with fonts for print use, suiting the limitations of onscreen display.^[5] Carter has described spacing as an area he particularly worked on during the design process.^[6]

Distinctive visual identifiable characteristics[edit]

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Characteristics of the typeface are:

Lower case

• there is a square dot over the letter i
• the lowercase j has a serif on top that protrudes left
• the a is double-story

Upper case

• the capital Q's tail is centered under the figure
• the uppercase J has a serif on the top that protrudes left
• the uppercase I has two serifs on the top and bottom

As an example of the approach of making similar characters easily distinguishable, the digit 1 (one) in Verdana was given a horizontal base and a hook in the upper left to distinguish it from lowercase l (L) and uppercase I (i). This is similar to the digit '1' found in Morris Fuller Benton's sans-serif typefaces News Gothic and Franklin Gothic.

Prevalence[edit]

Released in 1996, Verdana was bundled with subsequent versions of the Windowsoperating system, as well as their Office and Internet Explorer software on Windows, classic Mac OS, and Mac OS X. Since at least Mac OS X 10.4 it is even bundled with Mac OS itself.^[7] In addition, up until 2002^[8][9] it was available for download from Microsoft's web site as freeware ('.exe' files for Microsoft Windows and in '.sit.hqx' archives for Mac OS) under a proprietary license imposing some restrictions on usage and distribution, allowing it to be used by end users in any system supporting installation of 'exe' or '.sit.hqx' files and supporting TrueType fonts.^[10] The downloadable files are still available legally from third-party web sites; see the External links section. However, these files include only old versions of Verdana and updated versions are not available as a freeware.

According to one long-running survey, the availability of Verdana is 99.70% on Windows, 98.05% on computers running Mac OS, and 67.91% on free operating systems like Linux.^[11]

According to a study of online fonts by the Software Usability and Research Laboratory at Wichita State University, participants preferred Verdana to be the best overall font choice and it was also perceived as being among the most legible fonts.^[12] However, Microsoft's font manager Bill Hill wrote that 'with its large x-height and very generous spacing, it never felt comfortable as an eBook font'. He noted that Microsoft had commissioned an alternative version of the pre-existing typefaces Berling and Frutiger for its Microsoft Readere-book product.^[2] Despite this, Verdana was initially used as one of the bundled book-reading fonts on the iPad before an update in 2011.^[13]

Microsoft variants[edit]

Verdana Ref is a custom version of Verdana for use with Microsoft Reference. It is used in Microsoft Bookshelf 2000, Encarta Encyclopedia Deluxe 99, Encarta Virtual Globe 99, Office 2000 Premium, Publisher 2000. MS Reference Sans Serif is a derivative of Verdana Ref with bold and italic fonts. This font family is included with Microsoft Encarta.

Tahoma is similar to Verdana but with tighter letter spacing. The Windows Mobile core font Nina^[14] is a more condensed version of Tahoma and Verdana.^[15]

Verdana Pro[edit]

Microsoft licensed rights to Verdana to Font Bureau for a new Verdana Pro release, published in 2013. Verdana Pro adds correct German closing quotation marks, light, semi-bold and black styles with italics, as well as condensed styles with italics across all weights. The expanded family was designed for organisations which had made extensive use of Verdana due to its availability but desired additional versions for greater flexibility. It is sold separately through print and web licenses, being sold by Font Bureau and Ascender,^[16][17] although the Windows 10 users can acquire it free of charge from Microsoft Store^[18] or by activating the Pan-European Supplemental Fonts optional resource on the Settings app. A similar Georgia Pro release was announced at the same time.^[16]

Combining characters bug[edit]

In the past, Verdana (v. 2.43) had an incorrect position for combining diacritical marks, causing them to display on the following character instead of the preceding. This made it unsuitable for Unicode-encoded text such as Cyrillic or Greek. This bug did not usually reveal itself with Latin letters. This is because some font display engines substitute sequences of base character + combining character with a precomposed characterglyph.^[19] This bug was subsequently fixed in the version issued with Windows Vista. It is also fixed in Verdana version 5.01 font on Windows XP by installing the European Union Expansion Font Update from Microsoft.^[20]

Awards[edit]

In 2006, the Verdana typeface was named in the list of British design icons in the Great British Design Quest organised by the BBC and the Design Museum. Carter's typeface appeared on a list which included Concorde, Mini, Jaguar E-Type, Aston Martin DB5, Supermarine Spitfire, World Wide Web, London tube map, AEC Routemaster bus and the K2 telephone box.^[21]

Usage[edit]

In 2009, IKEA changed the typeface used in its catalog from Futura to Verdana, expressing a desire to unify its branding between print and web media. The controversy has been attributed to the perception of Verdana as a symbol of homogeneity in popular typography.^[22]Time magazine and the Associated Press ran articles on the controversy including a brief interview with an IKEA representative, focusing on the opinions of typographers and designers.^[23] Design and advertising industry-focused publications such as Business Week joined the fray of online posts. The branding critic blog Brand New was one of those using the Verdanagate name.^[22] The Australian online daily news site Crikey also published an article on the controversy.^[24]The Guardian ran an article asking 'Ikea is changing its font to Verdana—causing outrage among typomaniacs. Should the rest of us care? Absolutely.'^[25]The New York Times said the change to Verdana 'is so offensive to many because it seems like a slap at the principles of design by a company that has been hailed for its adherence to them.'^[26]

Characteristics of the typeface are:

Lower case

• there is a square dot over the letter i
• the lowercase j has a serif on top that protrudes left
• the a is double-story

Upper case

• the capital Q's tail is centered under the figure
• the uppercase J has a serif on the top that protrudes left
• the uppercase I has two serifs on the top and bottom

As an example of the approach of making similar characters easily distinguishable, the digit 1 (one) in Verdana was given a horizontal base and a hook in the upper left to distinguish it from lowercase l (L) and uppercase I (i). This is similar to the digit '1' found in Morris Fuller Benton's sans-serif typefaces News Gothic and Franklin Gothic.

Prevalence[edit]

Released in 1996, Verdana was bundled with subsequent versions of the Windowsoperating system, as well as their Office and Internet Explorer software on Windows, classic Mac OS, and Mac OS X. Since at least Mac OS X 10.4 it is even bundled with Mac OS itself.^[7] In addition, up until 2002^[8][9] it was available for download from Microsoft's web site as freeware ('.exe' files for Microsoft Windows and in '.sit.hqx' archives for Mac OS) under a proprietary license imposing some restrictions on usage and distribution, allowing it to be used by end users in any system supporting installation of 'exe' or '.sit.hqx' files and supporting TrueType fonts.^[10] The downloadable files are still available legally from third-party web sites; see the External links section. However, these files include only old versions of Verdana and updated versions are not available as a freeware.

According to one long-running survey, the availability of Verdana is 99.70% on Windows, 98.05% on computers running Mac OS, and 67.91% on free operating systems like Linux.^[11]

According to a study of online fonts by the Software Usability and Research Laboratory at Wichita State University, participants preferred Verdana to be the best overall font choice and it was also perceived as being among the most legible fonts.^[12] However, Microsoft's font manager Bill Hill wrote that 'with its large x-height and very generous spacing, it never felt comfortable as an eBook font'. He noted that Microsoft had commissioned an alternative version of the pre-existing typefaces Berling and Frutiger for its Microsoft Readere-book product.^[2] Despite this, Verdana was initially used as one of the bundled book-reading fonts on the iPad before an update in 2011.^[13]

Microsoft variants[edit]

Verdana Ref is a custom version of Verdana for use with Microsoft Reference. It is used in Microsoft Bookshelf 2000, Encarta Encyclopedia Deluxe 99, Encarta Virtual Globe 99, Office 2000 Premium, Publisher 2000. MS Reference Sans Serif is a derivative of Verdana Ref with bold and italic fonts. This font family is included with Microsoft Encarta.

Tahoma is similar to Verdana but with tighter letter spacing. The Windows Mobile core font Nina^[14] is a more condensed version of Tahoma and Verdana.^[15]

Verdana Pro[edit]

Microsoft licensed rights to Verdana to Font Bureau for a new Verdana Pro release, published in 2013. Verdana Pro adds correct German closing quotation marks, light, semi-bold and black styles with italics, as well as condensed styles with italics across all weights. The expanded family was designed for organisations which had made extensive use of Verdana due to its availability but desired additional versions for greater flexibility. It is sold separately through print and web licenses, being sold by Font Bureau and Ascender,^[16][17] although the Windows 10 users can acquire it free of charge from Microsoft Store^[18] or by activating the Pan-European Supplemental Fonts optional resource on the Settings app. A similar Georgia Pro release was announced at the same time.^[16]

Combining characters bug[edit]

In the past, Verdana (v. 2.43) had an incorrect position for combining diacritical marks, causing them to display on the following character instead of the preceding. This made it unsuitable for Unicode-encoded text such as Cyrillic or Greek. This bug did not usually reveal itself with Latin letters. This is because some font display engines substitute sequences of base character + combining character with a precomposed characterglyph.^[19] This bug was subsequently fixed in the version issued with Windows Vista. It is also fixed in Verdana version 5.01 font on Windows XP by installing the European Union Expansion Font Update from Microsoft.^[20]

Awards[edit]

In 2006, the Verdana typeface was named in the list of British design icons in the Great British Design Quest organised by the BBC and the Design Museum. Carter's typeface appeared on a list which included Concorde, Mini, Jaguar E-Type, Aston Martin DB5, Supermarine Spitfire, World Wide Web, London tube map, AEC Routemaster bus and the K2 telephone box.^[21]

Usage[edit]

In 2009, IKEA changed the typeface used in its catalog from Futura to Verdana, expressing a desire to unify its branding between print and web media. The controversy has been attributed to the perception of Verdana as a symbol of homogeneity in popular typography.^[22]Time magazine and the Associated Press ran articles on the controversy including a brief interview with an IKEA representative, focusing on the opinions of typographers and designers.^[23] Design and advertising industry-focused publications such as Business Week joined the fray of online posts. The branding critic blog Brand New was one of those using the Verdanagate name.^[22] The Australian online daily news site Crikey also published an article on the controversy.^[24]The Guardian ran an article asking 'Ikea is changing its font to Verdana—causing outrage among typomaniacs. Should the rest of us care? Absolutely.'^[25]The New York Times said the change to Verdana 'is so offensive to many because it seems like a slap at the principles of design by a company that has been hailed for its adherence to them.'^[26]

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Carter addressed this controversy during an interview in 2013:

Ever since there was that big ruckus about the IKEA catalog changing from Futura to Verdana, which I had nothing to do with and didn't even know about, people ask me about that everywhere I go. I give a talk about something historical and then at the end someone will get up and say: 'I started a petition to go back to Futura. You're a villain!' You get blamed for something you had nothing to do with.
There's a strange misunderstanding. A friendly guy came up to me at a conference recently and said: I signed that petition to go back to Futura. So I asked: what caused you to do that? And he said, well, Verdana is a screen font. You mustn't use it in print. So I said: OK, well, so you open the IKEA catalog, it's set in Verdana, with the big prices and everything… how do you tell it's a screen font? What is it about Verdana that says: this is a screen font? He had no idea. He just knew it because he'd been told. There are many people who make judgments without really understanding what the typographic issues are. Students are interesting—they'll say things to me like: my professor told me I cannot use Verdana and Georgia in print because they're screen fonts, but I tried it and it looks perfectly all right. And I can only say: Thank you! Go ahead!^[5]

In 2019, with its logo refresh, IKEA again changed their corporate typeface, from Verdana to a customized version of Noto Sans under the name Noto IKEA and is used in pair with standard versions of Noto Sans.

See also[edit]

Notes[edit]

1. ^Re, Margaret, ed. (2003). Typographically Speaking: The Art of Matthew Carter (2. ed.). New York: Princeton Architectural. pp. 41–2. ISBN9781568984278. Retrieved 26 March 2016.
2. ^ ^abHill, Bill. 'Apple updates iBooks with new book fonts'. Bill Hill 49 (blog). Retrieved 26 March 2016.
3. ^'Interview with Virginia Howlett, mother of Verdana'. Dmxzone.com. 2004-06-24. Retrieved 2013-09-21.
4. ^Will-Harris, Daniel (2003). 'Georgia & Verdana - Typefaces designed for the screen (finally)'. Archived from the original on 28 August 2013. Retrieved 9 August 2013.
5. ^ ^abMiddendorp, Jan; Carter, Matthew (October 2013). 'Matthew Carter'. Creative Characters. MyFonts by Monotype. Retrieved September 29, 2018.
6. ^Drucker, Johanna (2003). 'Typographic Intelligence'. In Re, Margaret (ed.). Typographically Speaking: The Art of Matthew Carter (2. ed.). New York: Princeton Architectural. p. 12. ISBN9781568984278. Retrieved 26 March 2016.
7. ^'Apple.com - Mac OS X 10.4: Fonts list'. Support.apple.com. 2011-11-06. Retrieved 2013-09-21.^{[dead link]}
8. ^Mark Hachman (2002-08-14). 'Microsoft Withdraws Free Web Fonts'. ExtremeTech. Retrieved 2010-04-13.
9. ^Microsoft (2002-07-25). 'TrueType core fonts for the Web FAQ'. Retrieved 2010-04-13.^{[dead link]}
10. ^Microsoft (2001-12-28). 'TrueType core fonts for the Web EULA'. Retrieved 2010-04-13.^{[dead link]}
11. ^Code Style: Most common fonts for Windows, Mac and Linux, full font survey resultsArchived April 26, 2013, at the Wayback Machine
12. ^A Comparison of Popular Online Fonts: Which Size and Type is Best?^{[dead link]}
13. ^Peters, Yyves. 'Version 1.5 Improves Typography in iBooks on iPad and iPhone'. FontFeed. Archived from the original on 16 March 2016. Retrieved 26 March 2016.
14. ^'E-books: an InfoDesign-Café discussion about their usability potentials and problems'. Web.archive.org. 2008-02-24. Archived from the original on February 24, 2008. Retrieved 2013-09-21.
15. ^'When in doubt, use Verdana'. Microsoft.com. Retrieved 2013-09-21.
16. ^ ^abColes, Stephen. 'Verdana Pro'. Font Bureau. Retrieved 28 July 2015.
17. ^Zeldman, L. Jeffrey (2010-04-18). 'Verdana Pro (and Con)'. Zeldman.com. Retrieved 2013-09-21.
18. ^'Verdana Pro'. Microsoft Store. Microsoft. 13 December 2017. Retrieved 1 December 2018.
19. ^'Underdots'. Tenser.typepad.com. Retrieved 2013-09-21.
20. ^'European Union Expansion Font Update'. Retrieved 2011-02-02.
21. ^'Long list unveiled for national vote on public's favourite example of Great British Design'. BBC. 18 November 2016.
22. ^ ^ab'Verdanagate,' by 'Armin' in 'Brand New' blog, August 31, 2009
23. ^The Font War: Ikea Fans Fume over Verdana. Time, (August 28, 2009)
24. ^Mel Campbell and Jeremy Wortsman, 'The Full Fonty: Why Type Nerds Went Mental Over Ikea', 'Crikey,' September 1, 2009
25. ^Verdana: Ikea's flat-pack font, Simon Garfield, The Guardian, 2 September 2009.
26. ^'Typography Fans Say Ikea Should Stick to Furniture', Edward Rothstein, The New York Times, September 4, 2009.

References[edit]

• 'Interview with Virginia Howlett, mother of Verdana' Retrieved September 9, 2005
• Friedl, Friedrich, Nicolaus Ott and Bernard Stein. Typography: An Encyclopedic Survey of Type Design and Techniques through History. Black Dog & Leventhal: 1998. ISBN1-57912-023-7.
• Macmillan, Neil. An A–Z of Type Designers. Yale University Press: 2006. ISBN0-300-11151-7.

External links[edit]

• Verdana font family (Microsoft typography)
• Verdana Pro font family (Microsoft typography)
• Verdana Ref font family (Microsoft typography)
• Downloadable version of Verdana (Core fonts for the Web)

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